To start with:
... I don't like this reimagining.
... I liked the original.
... I liked 3DS Ghost Recon-Shadow Wars (limited replay fun).
... I like Xenonauts (so far).
I think this game would be improved if you could NOT hire new soldiers.
They streamlined this game to play faster.
You want faster?
Start us with 14 soldiers.
Let us take ALL of them on every mission.
Then maybe I'll care about permadeath by the end of the game.
Then maybe there will be some strategy to the metagame.
Then maybe I'll name my soldiers.
This game is missing something that could take it from "(meh)okay" to "Great!".
This doesn't feel like it took years to make ...
... it barely 'looks' like it took years to make.
The original is still probably my all-time favourite game, but XCOM is extremely good; I've never really got the "either-or" mentality that liking a computer game precludes liking a different one...
My general (and somewhat lengthy) impressions so far now I've played for a couple of days.
Campaign: Evidently the part of the game that has changed most substantially. This has both good and bad points, and one's take on the latter is likely to determine whether you'll enjoy the game as much as the first one. The campaign isn't any more scripted than in the first game - as in UFO there are fixed techs you need to progress to the next story point, and you can only achieve them by completing certain objectives mostly involving capturing a live alien of type X. However, it is a lot more linear and structured; there's no freeform 'roleplaying' element to the campaign as there was in the first game. And if you choose not to (or are unable) to take on specific missions, you will often be heavily punished - while you get to choose which of several abduction sites, say, you focus on, the 'none of the above' option is essentially fatal. Similarly a terror mission must be undertaken or you will lose a country and likely a continent from the Council; it's very unforgiving if you lose a mission, and you can't, as I often did in the first game, ignore certain terror missions in the late game in favour of taking on more UFO missions for resources I needed. By the standards of any modern computer game I can think of, the campaign is long and presents rather flexible options, but if that's what the key appeal of the original game was to you, this won't meet your expectations. It's quite likely there is an optimal strategy that will always work, as is typically the case with these kinds of linear games - you aren't going to make fast progress by lucking out on a landed UFO early on or getting an early psionic alien capture (psionics enter the campaign at a scripted point in the game) that will affect your future strategy, or indeed be hampered by bad luck in the same way.
On the positive side, there are new mission types, a very nicely-designed psionic system that beats the old "You have psionic power, go knock yourself out mind-controlling everyone", and an interesting approach to the tech tree. This latter is actually rather large when considering techs, equipment, foundry upgrades and (the most poorly-developed) officer school. The new missions sadly don't integrate well with the story - you don't get any new story credit or even background info for escorting a VIP with vital information back to the Skyranger, for instance, and there should have been some randomisation of agent names or at least more than one of each type just to have some variety, rather than rescuing Anna Singh in China in one game and the UK the next.
Resources are in tighter demand than in previous X-COM games - elerium is needed for most later tech and units, most missions will be alien abductions rather than UFO attacks (expanding satellite coverage doesn't seem to do a lot to increase the number of UFO encounters) which don't provide it, and you aren't going to be getting it in quantities of 100+ anymore. Nor can you manufacture alien alloys, weapon fragments or corpses of various species, all of which are required for certain advances or production. Probably for this reason you no longer lose valuable manufactured equipment when your squad members die - this I'm less happy about, and I think there should be a game option to enable "lose kit when you lose a mission" for old-timers who want to play it that way (or anyone wanting more of a challenge, although this game is not 'easy mode'). Adding grades and related voiceovers to the Council briefings is a nice touch, but unfortunately doesn't seem to make any actual difference to the game - you don't lose funding for getting a bad rating or increase it for getting a good one, and while you're more likely to lose countries if you're not doing well, you aren't going to lose any you wouldn't lose anyway because the Council gives you an F.
Combat: This is where the new game really shines (as indeed you might hope). I've been playing on Normal, which is purportedly a level X-COM veterans should find a little easier than they're used to. In my experience, however, it starts out easier but rapidly becomes tougher - and I say this not from nostalgia but from playing UFO concurrently (on the only difficulty level originally available in that game, as my copy is unpatched). More importantly than straight difficulty, to get the results I get in UFO, I need to play in a far more sophisticated way in the new game. Not all missions will start with "pop smoke, leave Skyranger, methodically stroll down the map killing all the aliens". The new game's been criticised for limited numbers of maps, but differences in your spawning points, aliens' spawning points, number and unit mixes of aliens and mission damage to cover objects that's going to differ in each playthrough add a huge amount of replayability (though it remains unfortunate that, except with area weapons, you can't target cover objects - although you can target aliens in the knowledge that the shot will likely destroy cover whether it hits or misses, and that can be a reason for firing with a low chance of hitting).
Few of my missions (aside from the Council missions, which have a small selection of maps, spawn you in much the same place each time and are invariably populated solely by Thin Men even in the late game) play at all alike, a difference much more pronounced than in the original. One mission I'll drop straight into a firefight with half a dozen Mutons and Berserkers where I pop smoke and suppress enemies to take the flak off, in another I'll be luring Muton Elite out of their UFO to pick off one by one, making judicious use of retreats, flanking, overwatch, hunkering down and unit abilities; in yet another I'll be flanked by Heavy Floaters popping out of nowhere, destroy them in a firefight and advance, only to be forced to withdraw under cover of overwatching heavy plasma gunners as Chryssalids pour from the UFO before finishing (and in this case losing) the mission in a firefight inside the saucer itself.
The two-action system and the class system prompt some very careful planning - I'm now a real fan of the latter, although there are apparent balance issues at some levels (Squadsight is always the best option for a Sniper Corporal, and I try to promote Assault units heavily especially to get the 'free reaction shot within 4 tiles' promotion), and undoubtedly there will be 'best' build orders. I still think it's unfortunate that the stats of new recruits are identical.
The deliberately small squad size (and inability to pick up dead soldiers' weapons) makes it imperative to choose the right unit mix, and I should probably vary it by mission more than I do. Classes also work well with the new psionic system in determining who to train up - I'm undecided whether snipers should be trained preferentially so that they can use a psionic ability if forced to move, for instance, or assault units so that they have a long-range attack.
Flavourwise, for some reason I'm not finding my characters as memorable as names or getting as attached to them as I did in the original game (although I like the greater selection of random names, as well as the fact that - as a Starcraft veteran - I placed a soldier called Kerrigan in psi-training and she was the only one in that batch to be psionically gifted). I do like the new nickname system (especially when my heavy becomes Boom Boom - nicknames appear to be chosen from class-specific lists), probably because I'm focusing more on the abilities I've lost. And the ranking system does seem to mean you're unlikely to get the same types of heroic rookie stories as in the original game - your best-promoted guys are going to be the ones doing the killing. If you lose them you're often going to lose the mission; so far I've had only one 'desperate last stand' where I'd lost four members of the squad, one other was bleeding to death, and my plucky support squaddie killed the final aliens.
AI: I found the AI lacking on first encounter, however even on nominally Normal difficulty perhaps it gives you an easier time in the tutorial game, even after the tutorial stage? It's seemed pretty capable in my recent games - floaters now drop in cover and do a good job of flanking, grenades are thrown at appropriate junctures, and if you drop smoke against Mutons, chances are they'll respond by suppressing the unit in the cloud. Aliens can still be too stationary when in position regardless of context (so that they always won't move to escape being flanked), they tend not to move much from their spawning site until spotted (so, if you're up against a UFO's door in a firefight with Heavy Floaters outside, no one inside the ship will come to help or even to see what all the fuss is about), and melee units are easy to lure into traps - run away when you spot the Chrysalid, and it will always pursue. Which makes them much less frightening once you have weapons that kill them in one shot, since you can just set up overwatch. If you have a SHIV, if you retreat your people behind it, the Chrysalid will attack the vehicle it can't convert.