In the original trilogy, the Empire Officers are mostly older men.
Does this have to do with casting, or does the First Order have some reason for all of younger people in high positions?
First of all, let's dispense with the uber-obvious out-of-universe reason: Darth Mickey and their commercial strategy (including target demographics); and skip straight to the interesting in-universe stuff.
I would also like to point out that other top-flight commanders aren't always old: from out-of-universe Alexander the Great and his Companions; to in-universe, just-starting-Empire's 25 year old newly minted Darth Vader and reasonably young Tarkin.
It's hard to say "why" in general - other than my general impression from bits and pieces is that First Order isn't all that huge so the human resource pickings may be slimmish - BUT:
General Hux: According to Visual Dictionary (p.40), it seems that he got his position so young due to good old nepotism. His Daddy was a "highly placed official in the Imperial Academy... on Arkanis" (later in the same page it said the father was the "Academy commandant", and Hux's views and theories are extension of his fathers' training principles), so he got the job.
I seriously doubt that he qualifies on merits based on the details about him revealed so far (he thinks realistic simulations are enough for training - VD says "Hux's experiences in warfare are entirely theoretical". Don't see how that'd cause any issues). It's explicitly noted Kylo Ren has no respect for him "as a warrior".
You KNOW you ain't a great warlord when your harshest critic is an emotionally unstable perpertual teen who has his own backside kicked by a 19-year-old nobody scavenger from Bupkis, DesertPlanet (even if she has the genes and the Force - so did you, Kylo!)
Kylo Ren: he was the only Skywalker of an appropriate age available around with Daddy Issues, ready to be turned to the Dark Side. Beggars can't be choosers, so Emo Teen got the gig.
Captain Phasma: so far, no clear details have been revealed about her past to indicate what her path to power was. I'm not even sure of her in-universe age, although the actress is youngish looking.
Supreme Leader Snoke: He told Kylo Ren that he witnessed Galactic Empire Rise - which was 50+ years ago, so he's much older than that.
One more relevant piece of info that may explain this: several times in canon it's described how First Order is meaningfully different from Empire ideologically. E.g. they care about their TIE pilots. They make military hardware based on performance, not bureaucratic BS (both these facts are from Incredible Cross-Sections pages on TIEs).
Basically, to me, First Order sounds like a small, nimble, disruptive Silicon Valley startup.
As such, it's quite possible - though not yet shown in any canon - that Snoke chose to NOT include any old fogeys from the Empire, and do the startup thing with young and vigorous leaders. Wiping the slate fresh, so to speak. This is just a guess and may very well be wrong.
However, please remember that we have very little in-depth knowledge about 30ABY Galaxy far far away. (Just like ANH only showed us very little about either the Galaxy or the Empire, if you recall what we saw: 1 neutral Rim planet; Alderaan, 1 Rebel moon, 1 DeathStar, 1 mention of Galactic Senate and Sector Governors and that the Emperor exists).
For example, we don't know what happened to Admiral Rae Sloane, and her mysterious Admiral-Thrawn-like superior who showed up at the end of "Aftermath" novel. I can't possibly believe Disney/LSG released that without any intent to tie it into upcoming films.
 First Order isn't all that huge - proof.
We can guess at that from the novelization scene, where Hux asks Snoke to wait a couple of hours to pinpoint D'Quar to destroy it - as opposed to right-away destroying the whole Ileeesium system the planet's in. His reasoning: after destroying Hosnian System, First Order would benefit from including the rest of Ileesium's planets.
Hux was clearly gratified to be the bearer of good news. “We were able to track their reconnaissance ship back to the Ileenium system. We are coordinating with our own reconnaissance craft in the area in order to lock down the specific location of their base.”
Snoke replied with cold satisfaction. “We do not need it. Prepare the weapon. Destroy their system.”
Collected and composed as he was, Hux was not immune to surprise. “The system? Supreme Leader, according to the most recent galographics, at least two and possibly three habitable worlds circle Ileenium. Following the destruction of the Hosnian worlds, would it not be worthwhile simply to destroy their base and claim the remainder for the Order? We will have the location of the base within a matter of hours and—”
What??? Old Galactic Empire consisted of millions of planets. Debating the need for a couple of planets of 1 system is... rounding error of rounding error to them. They blew up Alderaan just for a demo. This clearly shows First Order is small and in need of resources.
To put it simply, the First Order was firmed pretty recently. Most of the people who joined were raised by Imperials, so they most likely would be young.
In addition to DVK-in-exile's excellent answer that focuses on some characters, there is a new more general answer from the recent novel Aftermath: Life Debt by Chuck Wendig.
The Shadow Council (which will certainly lead to the creation of the First Order) has big plans for children, basically taking people very young to make them brainwashed loyal soldiers. Brendol Hux, father of General Hux from The Force Awakens, is a member of the Council.
I can't find any quote from the book right now - I haven't read it myself yet. Anyway you can easily find many articles that mention this and reveal everything is in the book. Like this one, which I would use as a source (obviously the link contains major spoilers): http://www.gizmodo.com.au/2016/07/all-the-secrets-of-the-star-wars-galaxy-revealed-in-aftermath-life-debt/