I have the fortune of owning both Final Mixes, so I will share my experiences with them in this article.
Please be aware, if you have not yet played the original titles, there may be spoilers ahead.
Kingdom Hearts II: Final Mix starts off as the original did. It's not until later in the prologue you notice one or two extra scenes. Eventually, as you progress through the game, more awesome stuff is revealed, and once you reach the end, there are about 20 extra bosses waiting.
The bosses are perhaps the highlight of KH2FM. One special mention is to a boss that was intended for the original non-FM version: In The World That Never Was, when Sora fights Roxas - you actually fight Roxas! In my experience, he was the most difficult storyline boss in the game (with the exception of, perhaps, Demyx), but it is a memorable fight. If you play it right, you can also allow Sora to wield three - THREE - keyblades. At the same time. (Also, beating Roxas unlocks one of my favourite abilities in the game and a new keyblade. I won't spoil.)
There is also a secret area in Radiant Garden. Not just a single room - a completely new area! This area took me a long time to complete, as it requires near-mastery of most of the drive forms. You will need abilities such as Final Form's Glide and Valor Form's High Jump just to progress. The area also features brand new heartless. Unfortunately they are just palette swaps of other models, but are much harder in difficulty. And, if you make it to the end of the area, you are able to fight data versions of all 13 members of the Organisation - much harder than the originals, but very rewarding. For the members who debuted in Chain of Memories, there are also some flashy new Reaction Commands.
Another highlight of the game is the music from Yoko Shimomura - the new tracks are superb. With the remastering of 1.5's music, the same is likely to happen to II.5, and this will be totally awesome. I can't see it not happening and will be sorely disappointed if it doesn't.
One unfortunate aspect of the many new cutscenes is that, in the original Japanese Final Mix, they weren't voiced. To portray dialogue, there were only subtitles (and in Japanese, too, so I really had no idea what was going on.) This was a bit jarring, and, to me, kind of made the game feel a little less complete than if they'd left them out altogether. Some of these cutscenes made it into Days, so, with luck, the rest of them become voiced and make it to HD ReMIX.
There are also a lot of other features that you'll encounter over the course of the game - crown collecting like in BBS, new heartless, including mushrooms based on the Organisation, new keyblades, and a new drive form based on Sora's Kingdom Hearts I abilities. Oh, and Dodge Roll. You can use Dodge Roll now.
Birth By Sleep Final Mix has a little less content, and most of it is at the end of the game, unfortunately. However, once the game is finished, you are treated to a secret, playable, final episode - playing as Aqua, wielding the Master Keeper. There are still many unanswered questions about this episode - which will, inevitably, be answered in Kingdom Hearts III.
Overall, there is a LOT of extra content in these two titles. In addition, we can expect remastered music, HD graphics, and, of course, the Coded cinematics. Who knows - if we're lucky, we may even have a sneak peek at Kingdom Hearts 3!