I've always been told that powershifting is when you keep your foot planted on the gas and dont use the clutch, just "power" the shifter into the next gear. Its the way the old muscle car guys used to shift. Not at all good for the trans.
Speedshifting is as stated above when you just tap the clutch just long enough to grab the next gear, without lifting off the gas. Still sort of bad for the trans, but not as bad as powershifting. I have speedshifted my car a few times, but I try not to make a habit of it
