When should I use SATA 6gb/s?
A hard drive benefits from being connected to a 6Gbps SATA port when the drive is capable of sustained transfers faster than a 3Gbps SATA port support.
As you've surmised the ST3000DM001 drive can't saturate a 3Gbps port based on its sustained read figure (210MB/sec => ~ 1.7Gbps) so any benefit you would get would be limited to burst transfers out of the cache -- I would connect it to a 3Gbps port.
Spinning disk has a hard time getting up to 6Gbps speeds - In practice as of April 2012 only a few high-end 15K SAS drives can come close to 6Gbps performance (e.g. the IBM 44W2244 clocking in at 4.7Gbps), and the only drives that can saturate 6Gbps SAS or SATA connections are solid-state disks.
There are some advantages to a faster bus, even if a single spinning drive can't sustain that bandwith:
faster drives. Both higher-density magnetic drives, high-end SSDs and hopefully memristor-based drives can perform at high enough bandwidths as to make the faster bus a need.
big caches. some manufacturers slap a flash device on the drive, to act as a transparent write-through cache, this can be much bigger than the usual RAM cache, so the drive can sustain faster speeds a little longer.
drive arrays. Not too common with SATA connections, but there are some external eSATA enclosures, or maybe...
...not SATA but SAS. Since SAS and SATA share the same electrical and low-level specifications, improvements on one are quickly transferred to the other technology. Not only SAS drives are available on higher RPMs, but it's also very common to use SAS as a multi-device bus, which leads to the next...
...SATA multipliers/switches. Increasingly common, these devices allow you to use a big number of drives and external boxes on a small number of channels. It's almost as SAS!
The difference between these would be minimal because a hard disk's transfer rate is lower than 3GB/s.