On the same disk, none at all.
... in most scenarios.
If you have a database that has a very high amount of table allocations and deallocations, you can get some contention on the GAM, SGAM or PFS pages. By using multiple files, you spread out the load and reduce that contention.
In practice, this usage pattern rarely happens in user database. It is however quite common in tempdb. That is the reason why most best practises guidelines tell you that you need to create multiple data files for tempdb (allthough opinions differ on what the best rule of thumb for the number of files is).
(Also note that using multiple files for tempdb is best combined with (a) making sure that all those files are the same size and have the same autogrow setting, and (b) enabling trace flag 1118 to ensure that they remain equal-sized. You might also want to enable trace flag 1117 to further optimze tempdb usage. Google for the details).
On different disks, maybe. Are you seeing IO contention? If so, is the 'different disk' really different hardware? If yes to both, then you may get performance improvements.
Absolutely agree here. If the database is constantly hitting the same file hard, then you can gain poerformance by ensuring that the same data is spread out over multiple physical devices.