Is my favourite. A wide variety of characters, classes, monsters and weaponry. Solid mechanics (though not without some flaws and nuances that you will get used to). Branching story based on character alignment (lawful, chaotic, neutral). Doesn't have the long animations FFT suffers from.
Chariot System allows you to "undo" an action. The game stores your last 50 actions per battle allowing you to retry different strategies if things goes terribly wrong.
World System. Essentially rather than starting a NG+ you can return to certain "anchor points" in the story. Characters, equipment and level carry over, it is sort of like time travelling. You can re-attempt recruitment quests for missed characters or choose different story paths.
Art style is great. Music is fantastic.
FE vs FFT specifically though...
I prefer the story in FFT. It also offers greater freedom in terms of unit classing. It allows you to experiment a bit more while FE's units are pretty much dependent on characters. Each character only has a few to choose from if any at all (older FEs have less choice). FE has other things going for it though. Superior sprites and animations. Supports and character endings if you like that sort of thing.
Fire Emblem has the weapon triangle which I like. Lance > sword > axe > lance. One thing I didn't really like however is that luck plays a small to medium part in the game. You can really like a character and spend some time training him/her but if the game decides that you get crap at a level up repeatedly this unit will not be great.
FE has more games to its name and a number of them are good. FFT has 2 successors that are quite different from the original style. I didn't mind FFT Advance series but it is not as good as FE nor the original FFT in my opinion.
I recommend both, but if you must start with one then I say try FE first. It is generally shorter and in some ways simpler due to less options for character classing and such. Sacred Stones is my favourite in case you want a starting point.
Honourable Mentions
- Valkyrie Profile: Covenant of the Plume
- Jeanne d'Arc
- Disgaea series
- Wild Arms series.