I know that Vulcans and Romulan come from the same parent race. But beyond that, how did they branch off? For how long they have been apart? Are they more different than human and chimps? Can they breed?
They are still the same, biologically, and can interbreed. This means nothing in the Star Trek universe, as pretty much anything can breed with anything.
Romulans have several differences from Vulcans. They are subtle, but enough to prevent Crusher from treating Patahk successfully with a technique that works on Vulcans. In that episode it was discovered that Romulans actually have a high degree of compatability with Klingons, enough that a Klingon treatment worked for the Romulans where the Vulcan treatment failed.
Edit: To answer your other question, they are likely more closely related than humans and chimps, though this can't be guaranteed (radically different environments are one of the biggest causes for rapid evolutionary change, though this may be mitigated by a heavily technological society).
Whilst her history was never fleshed out canonically, Saavik was described to be half vulcan, half romulan, confirming that inter-breeding is possible.
According to the novels and comics, Saavik was born on Hellguard, an abandoned Romulan colony. She is half Vulcan and half Romulan. (A line of dialogue that would have revealed this in The Wrath of Khan was edited out prior to the film's release and never restored; as a result, the canonicity of this piece of information has been debated for more than two decades. It does, however, explain her somewhat emotional behavior in that film, though she adopted a more proper Vulcan demeanor in later appearances.)
(As pointed out in another answer, Spock is the product of a human/vulcan relationship and if that's possible, it's logical [sorry - couldn't resist!] to suggest that inter-breeding between two much more closely related species is possible. In fact, they're more "tribes" than species.)
The separation of Romulans was, if memory served, caused by those who didn't want to follow the teachings of Surak deciding to leave Vulcan, eventually arriving on Romulus (and Remus).
Well, as to the breeding question, it has been established that both Vulcans and Romulans can cross-breed with humans (Spock is the product of Human/Vulcan and Tasha Yar had a daughter by a Romulan). And it would be reasonable to assume Humans are less close to either race than they are to each other, therefore I'd think they should certainly be able to cross-breed Vulcan <> Romulan.
While the two species split off only 2000 years ago "as of the TNG era", they have many genetic differences. For example, the foreheads are often different, and many diseases have different effects.
While under normal conditions, they would be near identical, the Romulans were founded by only a limited offshoot population of the Vulcans. After that time, they were extremely isolationist for 2000 years "Enterprise info", and as a result, contact with Romulans was extremely limited, to the point that most Federation people had no clue what they even looked like. Based on this, I'd say that very few new members were added to the genetic pool. On top of that, the star that Romulus orbits underwent a supernova, so it's a pretty safe bet that prior to that incident, it churned out an above-average amount of radiation for a long time, which means a higher mutation rate. So we have an extremely isolated population with an increased mutation rate, so it's a safe bet that evolution would be slightly accelerated in this case. Based on that, it's possible that the difference between the two species could be approximately that of humans and neanderthals.
Its never fully adressed and the addition of the green and brow ridged Romulans and the taper off of all the green skin on vulcans just complicates things. orginally in tos you get the impression that Vulcans are greenish and Romulans are brownish making it even more confusing. My best guess is they are more like homo-sapian Idaltu compaired to homo-sapian sapian a only slightly less evolved subspieces or one that had better adapted to vulcan depending if vulcans are native to vulcan or ancient colonists of a now lost race. this would make romulans a little more then a racial group like almost if not a subspecies which would explain it.
Adding to the above, our present understanding of biology includes "Haldane's Rule", which says roughly that for species with sex chromosomes, the gender with the "odd" (in practice, truncated) chromosome is the one that gets hammered worse in interspecies crossbreeds: Less chance for survival (before and/or after birth), and infertility. For Earthly mammals, the affected sex is males with their Y chromosome. This is particularly visible among felines, with crossbreeds such as ligers.
So one would expect that male hybrids among the STU humanoids would be less common than female (due to fetal and early deaths), and likely infertile. On the other hand, much of this could be handwaved by futuristic medicine.
In "The Chase" we learn that many of the Alpha Quadrant species are related to a proto-humanoid species that seeded various planets with genetic material. These species included the Romulans, Klingons, Humans, Kardassians and we can assume others (Vulcans). It would stand to reason the genetic similarities amongst these species would be greater than species from outside of this population. Therefore, Romulans would share a greater family resemblance with Vulcans, but it would not be overly bizarre to have crossbreeding between any of these species.
Recall in the STOS episode -Return To Tomorrow- Sargon speaks of Henock as being "from the other side" in the contest that devastated their planet; Spock spoke of certain aspects of Vulcan prehistory being explainable as ancient colonization by explorers from Sargon's planet.
Perhaps there were two different species or subspecies making these colonizations together; the more agressive "Othersiders" and the peaceful "Sargonites." The "Sargonites" interbred with some protovulcans, the "Othersiders" with others, starting a diverging genetic line developing different features and prone to a more marshal philosophy.
This culminated in the more violent tribes breaking away from Vulcan during The Time of the Awakening and ultimately founding Roumuls.