In Voyager, the crew briefly enter "slip stream" travel in two episodes, with the second removing 10 years off their voyage home.
How does slip stream compare to, say warp 9?
Quantum slipstream transcends the normal warp barrier by penetrating the quantum barrier with a focused quantum field. What is the quantum barrier you say? It's technobabble, and it's not explained in canon.
The net effect is that quantum slipstream technology far exceeds the speeds capable with a normal warp drive, and rivals the Borg transwarp technology. Transwarp isn't a specific type of drive, but a class of propulsion technologies that exceed the normal warp limits. In this sense, quantum slipstream would be a subset of transwarp technology.
In terms of speed comparisons, it's very hard to make a direct comparison because the writers played fast and loose with the velocities each warp factor corresponded to.
Let's start with a given: based on the episode "Hope and Fear", Voyager was able to travel 300 light years with one hour of use of the quantum slipstream drive it had.
Under the original series warp scale, traveling that distance at Warp 9 would take about 5 months.
They revised the scale for The Next Generation, and it's assumed that's what all subsequent series used. The new scale is discussed in the technical manual for The Next Generation, and is reportedly based on the formula:
On that scale, traveling that distance at Warp 9 would take about 2.3 months. This should be considered the "canon" answer, as the formula isn't explicitly contradicted by any later canon sources.
However, in "Caretaker", it's established Voyager would take 75 years at maximum speeds to reach Earth, which is 70,000 light years away. This would indicate, based on the scale for The Next Generation, Voyager's top speed to only be around Warp 7.78. This is contradicted over and over again throughout the series as Warp 9 is routinely mentioned in dialogue.
Indeed, as Memory Alpha notes, the unpublished (and consequently non-canon) technical manual for Voyager reportedly stated the "maximum speeds" talked about in "Caretaker" was actually Warp 9.6, which would indicate a different scale than that which is used in The Next Generation.
But let's discount the estimate in "Caretaker" as being too slow and continue to assume the warp scale factor hadn't changed in Voyager's time (a reasonable thing to assume since Voyager and the latter part of TNG's exploits occur simultaneously).
According to the technical manual for The Next Generation, Warp 9.6 was only sustainable on a Galaxy-class starship for no more than 12 hours. However, Voyager—an Intrepid-class starship—was designed and constructed several years after the Galaxy-class starships. We're also given every indication that Voyager is a marked improvement over previous ship designs, so it's reasonable to posit that Voyager might be indeed be able to sustain Warp 9.6 as a maximum speed, in the same vein that Janeway described it in "Caretaker".
Based on this, we can conclude that under maximum speeds, Voyager would take a little less than 2 months to reach the same distance it traveled with just one hour of quantum slipstream use.
Of course as you mentioned, in "Timeless", the claim is made that 10 years was shaved off the journey. This is where you just need to take a step back and just ignore the math. Assuming this is true, and assuming the figures for "Hope and Fear" are correct, it would mean Voyager traveled 18,900 light years.
But, wait a minute: this is crazy. "Caretaker" establishes Voyager was at most 70,000 light years from Earth, but Janeway claims they only took 10 years off their trip. Which would mean Earth was actually more than 142,000 light years away in "Caretaker".
To put that number into perspective: the diameter of the Milky Way is only 100,000 light years. Yeah, this is not going to add up.
So we're left with a paradox. To resolve it, either the numbers in "Hope and Fear" are wrong, the numbers in "Timeless" are wrong, or the quantum slipstream drive in "Timeless" is significantly slower than the one in "Hope and Fear".
For the sake of argument, let's say the one in "Timeless" is a "baby" quantum slipstream drive, and is just nowhere near the speed of the original version. Given it shaved 10 years off the trip, that means it traveled 9,333 light years. What would that tell us about Voyager's cruising speed?
Well, it tells us that it would take 10 years to travel 9,333 light years only if you were going warp 7.78, which is vastly lower than regular warp limits. I'm pretty sure they could've went faster if they got out and pushed.
And that's exactly the same estimate in "Caretaker", isn't it? Assuming Voyager's top speed isn't around warp 7 or 8, it seems Voyager is running off of a different warp scale from The Next Generation that's internally consistent, at least with these two data points. And if it is, it means Voyager was just way slower than earlier ships established in The Next Generation and Deep Space Nine.
My head's starting to hurt, but let's just continue to use The Next Generation's warp scale and compare this cruising speed to the distance mentioned in "Hope and Fear": 300 light years. To travel that distance at warp 7.78, it would take a little under 4 months.
The lesson to be learned from all of this is that the only concrete thing about quantum slipstream technology is that it's really fast and served as a fairly decent plot device to back out of a really huge and unreasonable time estimate established in the first episode.
As an aside, I point out the problems with the "Caretaker" estimate to illustrate the difficulties of direct speed comparisons, but as pointed out in the comments, it could be that Janeway really meant "maximum safe cruising speed" instead of "maximum speed", which would indicate she was unwilling to unreasonably tax Voyager's engines just to gun it back home.
In this case, the only discrepancy in warp scale factors would be the figure in the unpublished and non-canon technical manual for Voyager, and one starts to wonder if those sorts of technical discrepancies is why it was never published.
In Caretaker, Janeway states that they are 75 years away from home at maximum warp. Since they start out about 75,000 light years from home, this presumes that Voyager's maximum warp is about 1,000 light years per year, or just under three a day. This is backed up by the episode Prime Factors, where Torres, the chief engineer says, "40,000 light years...that would knock about four decades off our trip."
Using the slipstream drive, they shave ten years off their journey, which means they traveled about ten-thousand light years in about half the length of the episode. Assuming this is in real-time, that's about five hundred light years a minute.
If you ignore the other contradictions in the series(and there are a lot of them), then slipstream velocity is thousands of times faster than warp drive. However, there are contradictions, such as:
1) In Scorpion Pt. II, Chakotay says that 40 light-years is a five-day journey at maximum warp, or 8 light-years a day. That's about three times faster than stated above, which would cut the travel time back to the Alpha Quadrant to about 22 years.
Bottom line: Slipstream velocity is much faster than warp velocity. By how much, depends on which episode you watch.
Slipstream is similar to transwarp in that it creates a conduit in deep subspace. Energy is channelled to the deflector instead of to the engine field coils. Warp drive is a completely different technology, that creates a closed bubble of normal space and pushes it through subspace.
Think of warp drive like pushing a bubble through treacle, and slipstream like rolling a ball through a pipe. Each ship would be vulnerable to the stresses of another propulsion method.
Also, voyager's deflector is designed to move space debris at the atomic level, quantum slipstream operates at the much smaller quantum level and manipulates a field.
for speed comparisons, warp 9 is about 1516 times the speed of light, about 1 lightyear every 6 hours. I calculate slipstream at 60,000 lightyears in 3 months, (666/day) to be 166 times faster (Voy: Hope and Fear) . This could be variable, and there could be different speed factors as in warp travel.
when Chakotay is trapped on the dauntless it travels 15 lightyears in about 5 minutes, giving 180 an hour, or 4300 a day; a thousand times faster than warp 9. voyager travels 300 lightyears before the conduit gives out about 7200/day or 2000 times faster than warp 9.
As a good rule of thumb I would say it is about 1000 times faster than warp 9, around warp 9.9999+.
As far as I recall, the first version of Quantum slipstream from 'Hope and Fear' was stated to be able to traverse 300 Ly's in 1 hour (Voyager couldn't sustain it longer due to Quantum stresses as Paris indicated). That amounts to 7800 Ly's per day, or 7 and a half days to go through 60 000 Ly's (at the time). This could indicate that the fabricated SF message was touting a much more reduced speed that would be 'safer' for long duration, although I cannot imagine why since the instances in which QS was used seemed to have conformed to 300 Ly's per hour - or 5 Ly's for every minute. This could be a simple mistake by the writers (we do know they were quite bad with consistency, though admittedly, this particular example could mean a maximum sustainable cruise velocity at Slipstream amounting to 20 000 Ly's per month, or roughly 714 Ly's per day, and about 27 Ly's per hour - about 11.11 times differential).
The version 2 of Quantum Slipstream as the crew devised it in 'Timeless' seemed to have been a lot faster. Per the dialogue in the series, it was stated that the phase variance problem doesn't start to arise until 17 seconds into the flight. That doesn't leave a lot of time for Voyager to be knocked out of the Slipstream tunnel and meet its icy demise (as indeed we've seen). I estimated that version 2 of QS as devised by the crew was able to cut through roughly 10 000 Ly's in about 1 minute (which seems consistent with how long Voyager was in Slipstream until after the trouble began and future Kim managed to correct the problem), maybe 2. If the demise was relaid to us in real-time, then 1 minute seems a bit more accurate.
10 000 Ly's per 1 minute does seem realistic mind you, since the QS drive version used a combination of different technologies (so it is quite possible that 7 was able to combine her knowledge of Transwarp to Slipstream). It had Borg technology in it, benamite Crystals, a Quantum matrix... Essentially, the outer shell of the Warp core was completely covered into a different configuration (as opposed to having a few additions like the modifications that were seen in 'Hope and Fear' to allow Voyager breaking through the Quantum barrier and catch up to the Dauntless).
If the 10 000 Ly's per 1 minute is accurate, that would amount to about 2000 times increase compared to the maximum of the first version (300 ly's per 1 hour - or 5 Ly's per 1 minute).
Compared to Voyager's maximum sustainable (diminished/damaged) warp capability of roughly 1000 Ly's per 1 year... that amounts to about 2.75 Ly's per day, or about 0.001 LY's per minute.
That would make v1 QS about 5 000 times faster than Warp 9 (9.975 would be as fast as Quantum slipstream as seen in 'Hope and Fear' if you factor in exponential increase in velocity past Warp 9.9 and that 9.9 = 21 473 times C).
QS v2 (if my calculations are close) would be roughly 10 million times faster than warp 9.
Doesn't anyone else think that if Voyager's actual sustainable cruise velocity was Warp 9.975 would be ridiculously slow in the 24th century? The Enterprise D estimate in Season 1 of TNG being able to traverse 2.75 million ly's in just over 300 years (which amounts to 46 Ly's per day - roughly) at Warp 9.6 no less.
And most of the series is consistent that Warp 8.5 or Warp 9 might be more akin to 1000 ly's per year.
Also, Voyager was never seen cruising at Warp 9.975... it was ALWAYS cruising at Warp 6 - this wouldn't make any sense for a ship capable of sustaining a velocity which is much higher. Which leads me to think that in-universe wise, Voyager likely suffered heavy damage as a result of its pull to the Delta Quadrant by the Caretaker (as was evidenced in the pilot) and prevented the ship from reaching anything past Warp 9.5 in a sustainable manner (meaning that it won't threaten to shake the ship apart) - and effectively, it 'euqalized' Voyager with other SF ships of its era (eliminating its superior Warp technology as a factor).
But, if you ask me, the writers would have been much better off at putting Voyager's Warp 9.975 as a sustainable cruise velocity and indeed allowing for 300 Ly's per hour... and, by those estimates, they should have pushed Voyager well beyond the Milky Way galaxy - about 197,100,000 Ly's away in total (to have Voyager take 75 years to get back at Warp 9.975).
That would have been a lot more interesting and showed proper/real advancement in Warp propulsion for the Federation. But since it didn't happen like that, its just as likely that Voyager was normally that fast, but with the damage it suffered, it was 'handicapped'