300mm is definitely hand holdable. As previously noted, you'll probably want around 1/500 shutter if you don't want to worry much about keeping yourself stable, which in lower light does mean jacking up the ISO to compensate. You can likely get away with 1/250-ish or even longer with practice at holding yourself still. If wildlife is the thing, you'll want the faster shutter speeds anyway to help freeze subject motion.
Having said all that, choosing between them does depend on what you want more. Do you need the 0.5x semi-macro of the Tamron? The Canon isn't far behind anyway at about 0.3x. IS does come in handy if the subject isn't moving quickly.
_________________Canon DSLRs: 7D2, 7D1, 5D2, 1D, 600D, 450D full spectrum, 300D IR mod
Lenses: EF 35/2, 85/1.8, 135/2+SF, 28-80 V, 70-300L, 100-400L, TS-E 24/3.5L, MP-E 65, EF-S 10-18, 15-85
3rd party: Zeiss 2/50 makro, Samyang 8mm fisheye, Sigma 50/1.4A, 150 macro, 120-300 f/2.8 OS
Compacts: Sony HX9V, Fuji X100.