3gb/s vs. 6gb/s is actually a pretty small visible difference. However, it depends a lot on what you plan on doing with your computer. The only scenario I can imagine justifying the need to purchase a whole new drive for the added speed is if you were a video producer, or anything where you're moving and/or altering large files on a regular basis.
tl;dr Don't bother spending more on a new drive unless you work as a video editor or something along those lines.
Video editing doesn't change anything. Mechanical drives can barely go faster then SATA 1.5. If your mechanical drive can only transfer at 135MB/s, then it doesn't matter if its on SATA1.5, 3.0, or 6.0.
As mentioned if you need speed, then get an SSD. Otherwise it doesn't really matter much.