To give an answer to your question, there have been several Trek Rpg's. Probably more accurate information than I can give here via a quick post available via a quick google or wiki search.
I believe there was one trek game prior to the FASA system, but I have no familiarity with it. Also, most of the RPGs listed have some kind of fan based web presence that one could look up to find add on materials (of various qualities, but some useable or even rivaling published material). Note I think even FASA managed to come out with a (first season) Star Trek: The Next Generation source book before losing their license.
The Star Trek rpg's I know about are:
Last Unicorn Games published three differently themed but system compatible Trek series sourcebooks, and a variety of supplements for each one before (wait for it) losing their license. Each series book was based on one of the series at the time, i.e. The original series, the next generation, and deep space nine.
from what I remember, the system was playable, but a bit crunchy. By today's standards far too rule and mechanic heavy for quick play. But that is my opinion. Steve Long, one of the games authors created an exceptional supplement to the game that makes the whole game worth looking at, if you are into starships and their statistics in game terms. The core was Stardock which added to the starship rules in the book, and a series of ship recognition manuals that covered ships from various factions.
I think there is quite a bit of supplemental material (fan based) for the game online with a little digging.
In my opinion, the gems of the entire system for anyone interested in gaming or writing trek are the three gamemaster's guides, one for each core book, that go into developing a game adventure/story that captures the feel of the shows. They really break things down into themes and almost TV like terms, even showing the three act structures etc.
Then Decipher acquired the license, and published their star trek rpg in a series of books starting with a players guide, a narrators guide, a Starfleet guide, and I think books on aliens, creatures, starships. There were two pdf only books that didn't make print in time. One is a worlds of the federation sourcebook, the other I think is a mirror uninverse sourcebook, but don't quote me on that, they are hard to find.
I think Decipher simply lost interest in supporting the line due to the low sales in comparison to their Card Game.
The Decipher version seems a bit more streamlined to actually play, but the books are poorly organized, which makes character creation and rules look up a bit of a chore. But they are pretty, and the actual character development system is interesting, as far as giving characters abilities within their specialty that mimic those shown by main characters on the shows.
There is also a GURPS version, based on the Task Force Games Starfleet battles universe (I won't go into the differences much. Kind of a more militaristic, original series era, with the non-bumpy Klingons an romulans at war with the federation. It's published by Amarillo Design Bureau. But I don't think they have ever gotten around to doing the starship combat part in RPG terms, despite it's origin as a tabletop starship combat game.
Finally, the newest entrant, the Modiphius Games Star Trek Adventures. It's currently in print, currently supported (although just starting), so has one Core book (used for running games in any era), and a book of adventures, and an ongoing "Living Campaign" that is free to download. Also some cool looking miniatures, and custom dice, and a planned line of geomorphic tiles, and more books in the pipeline.
It really is the most "feels like a Star Trek show" than any of the other RPGs because I think it was designed specifically with that in mind from the start. And the rules show that they are more interested in creating the feel of star trek, and the drama and story of that, than say a miniature combat tabletop game, or a more combat oriented RPG.
There is definitely a feel that solving non-combat problems in a dramatic, interactive way is a big part of the game, and I like that.
That's my short answer. I hope it helps.