Well, I did some experiments on FULLTEXT and the results were not like I would have wanted them to be. For instinct I had a hard time applying the index to varchar columns, even though it is documented that the index can be applied to varchar columns. I also couldn't find in the documentation how the match score was calculated ... seemed to me as a pretty random number with no roof or floor. It is a while since I was playing with those things so this feature might have been experimental at that time, so I might just be telling you the bollocks here 🙂
I think you should consider what your search is for when you choose how you're going to do your search. If you have a huge table you want to look up in, I would recommend using the FULLTEXT index. The difference between using REGEXP and LIKE is that one supports regular expressions and the other not. Come to think of it, REGEXP probably is a bit slower so using LIKE would probably be a more preferable choise.
-- Hope I didn't make things more complicated then they already were 🙂