The bigger concern with a SATA2-only computer is whether or not the BIOS supports AHCI mode. You need AHCI for NCQ (native command queuing) to work. The biggest speed increase from a SSD isn't sequential transfers, which is what SATA2 limits (300 MB/s vs 600 MB/s on SATA3). It's from random transfers of small files.
A modern HDD can hit about 150 MB/s on sequential transfers, so a SSD is only 2x-4x faster. But 4k random read/write speeds on a HDD are about 1.5 MB/s at best. On a SSD they're about 15-30 MB/s, which is 10x-20x faster (actually more since most HDDs are down around 0.8 MB/s). With NCQ, the SSD can queue up multiple random transfer requests (that's what the QD you see in some of the disk benchmarks are - queue depth) and complete them without needing further instructions from the computer. By doing this, SSDs can typically hit 150-300 MB/s at random 4k transfers with a QD=16 or 32. So at these types of tasks (e.g. reading lots of small files to load a game) they can be several hundred times faster than a HDD.
If you're running the SSD in SATA mode instead of AHCI, you lose NCQ and the random 4k read/write speeds will plod along at "only" 15-30 MB/s. The drive will "only" be 10x-20x faster than a HDD instead of hundreds of times faster.