A couple of considerations that come to mind for a database: (a) efficiency, and (b) dynamicity. The tree structure isn't particularly efficient (it has a lot of space overhead, and I think it's also not as fast), so you'd prefer something else if possible. So people often use suffix arrays + LCA preprocessing instead. However, those are hard to make dynamic (your database is constantly changing), so I'd expect that still makes it tough to use them in databases. Though I suspect this isn't the whole story.
Not _quite_ a suffix tree, but a similar structure called a GADDAG is sometimes used in word-generating board games (Scrabble, Words With Friends).
GADDAGs are different because they're more like "reversed prefix" trees, but they help with the issues of placing a new word from a dictionary onto the board using the existing letters of another word.
Every time I expect it to show up (e.g., for pattern matching in databases), I almost always see something else used instead (e.g., trigrams)