The '06 has more bullet weights, and is a little more efficient on powder usage. Factory ammo will likely be cheaper for the '06.
The 7mm mag bullets have a little better BC, and therefore shoot slightly flatter; retained energy at longer ranges will be slightly better with the 7mm Rem Mag with same weight bullets in each.
Muzzle velocities using either cartridge with same weight bullets will be nearly identical, with pressures slightly higher in the 7mm mag with most powders.
With most rifles, you'll get an extra cartridge in the magazine with an '06. If the same rifle carries 4 '06 cartridges in the mag, it'll only usually hold 3 if it's chambered for the 7mm mag.
If you handload, brass usually lasts longer for the '06.
I shoot a 7mm Rem mag, mostly because I've been shooting the cartridge since around 1986. Had I switched to an '06 at that point instead of a 7mm mag, I'd probably be using an '06 now.
Lest I sound like I'm favoring the '06, I'm not. I love the 7mm Rem mag cartridge, and it's my favorite all around big game cartridge for North America. Even so, an honest evaluation based on cost of shooting (factory or handloaded ammo), performance, and overall advantages will show the '06 to have an advantage for a lot of shooters.
Daryl