It does actually, though I assume it would be dependent on the software to some extent.
Since I didn't know how much of a difference it made (only that it did make a difference), I timed it.
Using Xilisoft Video Converter Ultimate. i5 2500k @ 4.4Ghz. AMD HD 6850. 8GB 1333 ram.
Four videos (43min-ish TV episodes) simultaneous (one per core). MKV > MP4 with default settings for sh!ts and giggles.
With AMD APP acceleration ON - 4min, 18sec (two of them were done in just under 2min)
With AMD APP acceleration OFF - 7min, 36sec (one was done about 45sec before the other three)
There's a toggle for activating CUDA as well (nVidia cards), which I can assume would give similar results.
EDIT - More info. Source files were about 220MB each. Resulting files were about 400MB. Yep, highly compressed originals.
It isn't really the quality of the video card imo, it's the extras: most newer cards have some sort of acceleration feature (APP or CUDA). Other things that help are CPU clock speed, number of cores and hyperthreading (which I don't have). Ram speed helps, but I'm not sure what kind of effect it has. I would, however, suggest a GPU with DDR5.