To expand on what Hawkeye22 said:
For HDDs, there is no advantage at all. The 6 GB/s refers to the SATA connection between the drive and the controller on the motherboard. That link can carry up to 6 GB/s. However, no HDD can transfer even 2 GB/s, so the 6 GB/s link is overkill. It will work fine, but it's not an advantage.
For SSDs, some of them can transfer faster than the 3 GB/s limit of the previous generation of SATA connections. So they would be slowed down by using SATA II with only 3 GB/s of capacity.
Some people have derided putting 6 GB/s links on HDDs as an advertising gimmick. I suspect that it is cheaper for a company to have all of their controllers be SATA III, rather than making, stocking, and keeping track of SATA II controllers separately for slower drives.
By the way, SATA II and SATA III are compatible. If you connect an SATA III drive to an SATA II controller, or an SATA II drive to an SATA III controller, you get a perfectly good link at the slower maximum speed of 3 GB/s.