Yes, different heads will have different inherent characteristics, as well as pre and power amps. Some are designed to be more transparent than others. Where the tube tone comes from depends on who you ask, I say it's the whole package. A little pre-amp distortion will be high and spicy, power tube distortion is where the real power and balls comes from IMO.
As for the cab you're using, I'm intrigued. As far as I'm aware the V4 never came with a tilt-back 4x12 - it was either a square 4x12 or a tilt-back folded horn 2x15 with the V4B. The V2 cab was a tall 4x12, but I don't know if those were ever made as a tilt-back or just the tall cab on a cart. It's possible, but I'm not 100%.
If you want more power out of those heads, get more speakers. Or get a new head with a similar tone but more power. Those a probably your best bets.