Not a whole lot of difference between the two. Both will handle the game you noted.
Depends on what you want. If you like short action rifles then go with the 308Win.
If you prefer a standard length action, take the 30-06. The short action bolt rifle will
probably be about 1/4 to 1/2lb. lighter, too. No real practical difference in speed of
bolt operation.
308Win. is considered the more accurate of the two, but for big game hunting purposes this really makes no difference. However, with varmints being considered, I'd opt for
all the accuracy I could likely get, and go with the 308Win.
Recoil is slightly less with the 308Win. Of course, this depends too on the weight of the rifle, and stock design.
The 308Win. will generally come within about 100fps of velocity you can get with the
30-06. Nothing to worry about there.
If you anticipate the need for using 200gr. or heavier bullets, then you need the 30-06.
But, if you can get by with 180gr., and you should be fine for the game noted, then the
308Win. will do okay.
Both are available in a wide variety of make/models of rifles, and ammo is readily available most anywhere you go.