I've done both. I like files for very simple problems and databases for complicated problems.
For file solutions, note these problems as the number of files increases:
1) Much more disk space is used than you might expect, because even tiny files use up a whole block. Blocks are fairly large on filesystems which support large drives.
2) Most filesystems get very slow when the number of files in a directory gets very large. My solution to this (assuming the names of the files are reasonably spread out across the alphabet) is to create a directory consisting of the first two letters of the filename. Thus, the file, "animal.txt" would be found at an/animal.txt. This works surprisingly well. If your filenames are not reasonable well-distributed across the alphabet, use some sort of hashing function to create the directories. Sounds a little crazy, but this can work very, very well, and I've used it for very fast solutions with tens of thousands of files.
But the file solutions really only fit sometimes. Unless you have a great reason to go with files, use a database.