In general, it seems to me that the answer to this question *for a Handi-rifle* is easy: the .280 rem, since the main advantage 7mm-08 has over 7x57 or .280 (in a repeating rifle) is the short action, so you can have a shorter, stiffer, lighter receiver. But in a single shot, that doesn't matter, so why not get the one with a little more oomph, in case you need it?
But, OTOH, as pointed out, if you're talking short barrels, you'd get only a smidge extra oomph from the .280 - but you would get SOME extra power, particularly in heavy bullet loadings. Even if you lopped off your .280 to 22", with a 175 gr bullet, you'd still get at least 100 fps more out of the 280 than the 7-08.
As for a good woods gun, that would be one choice, but a better one for thick woods with few and far between clearings (and small ones) would be a .45-70 or .35 Rem.