Thexder is a mecha who can transform into a jet, something which was obviously immediately appealing to gamers, especially computer folk who were probably getting sick of the usual adventure games. In mecha form, your laser will automatically target any foe as long as you're facing them, so there's no need to jump up and down to hit them. In jet form, the laser can only fire straight ahead, but it's necessary to travel through narrow spaces and reach higher areas. The levels consist of fairly small mazes consisting of many, many narrow corridors. It's one of those games that somewhat historically relevant - like Hydlide, it was popular during the day for being revolutionary, but the game itself hasn't exactly aged well. It's also brutally difficult.
There were two Thexder games created for Japanese computers - the original and its sequel, Firehawk. Both were ported to DOS and released in North America by Sierra, who also grabbed Silpheed, Sorcerian, and Zeliard. For the first game, the cool theme music didn't make the transition due to the awful PC speaker, and some of the special effects were removed, but otherwise it's a pretty alright port. The second game had its story cut out but actually improved the graphics a bit. Coincidentally, Square also ported the first game to the Famicom. It removed most of the stages and redesigned the ones that were left, so it's only the same four stages that repeat over and over - the original version has many more levels. Sierra also released a new Thexder game for Windows 95 somewhere around the mid-90s, which was dreadful, because the sprites were tiny, the action was choppy, and somehow the level design was even worse than ever.
Anyway, as near as I can tell, Thexder Neo is a remake of the original game. The first level is definitely almost exactly the same, although I can't say any more, because I never beat the first level of the original Thexder. (As I said, it's really hard!) This PSP version adjusts the difficulty level a bit, so it's much easier to handle. It controls much smoother too, and adds in an option to lock your direction, so you walk backwards while still firing forwards. Enemies drop various orbs, some of which will recharge your energy gauge, while others will expand its maximum capacity. You'll want to grab as many as these as possible, because not only does your energy drain when taking damage, but it'll also slowly drain as you fire. There's also a shield, although I never quite figured out its functionality nor how to recharge it.
Now, remaking games always brings up an interesting question - how faithful do you want to remain to the source material? The best, like Bionic Commando: Rearmed, take all of the good stuff and polish all of the bad stuff to make for a game that feels somewhat modern while still maintaining a retro sensibility. Thexder does not do this. The tight corridors and obnoxious enemies, many of which require several seconds of constant lasering to be killed, are still here. Navigating the mazes just isn't any fun, because, naturally, they're full of dead ends, and there's no real benefit to exploring. There's even some cases where you can get trapped and need to die to start the level over. The way to "fix" these problems would be to redesign the levels completely and rebalance the enemies further...but maybe it wouldn't be Thexder then? Game Arts already took the basic concept - the auto-aiming fire, anyway - and turned it into Alisia Dragoon for the Genesis, which would've been a much better game to remake. Of course, she doesn't carry quite the weight amongst retro nerds, and she can't transform into a jet, but she does have tits, which has to count for something.
Thexder Neo feels cheaply put together and not terribly interesting unless you're one of the few that actually has some nostalgia for this game - and if you're not Japanese, then you probably don't.