One of the big differences for me is:
Sphinx is the fastest but it doesnt work on Shared Hosts.
phpBB Native is maybe the one that will give better search results BUT if you've let say a large board, let say 300Mb of posts table can create a 600mb of search table. (It will depend the configuration on ACP too) But with this one I can search for words with 3, 2 and even 1 letter.
MySQL Fulltext just create an index at database so it go to 2mb maximum compared to the 600mb of phpBB native lol, but the downside is that it can be slower at give the results, but the real bad thing at least for me is that it is limited to words with 4 letters at minimum. I wanted to be able to search too words with 3 letters as "men" and even 2 letters (in my case), its crucial to find information on my board.
This MySQL limitation I guess could be fixed on dedicate server, but on share hosts you cant change the 4 letters limitation
About PostgreSQL this is if I used that kind of sql but I use MySQL instead, so I never new about this neither how it works.