The 270 is the the 30-06 high performance brother. The 06 is a military round pushed into hunting. The 270 was specifacally designed hunting cartridge to improve the ballistics of the older 30-06 that is under bored and was already in military service.
There is nothing a 06 can do that a 270 can't except the 270 will reach out further doing it.
For NA species, either will be great. If you are just hunting deer a 243 is a great choice. If you want a 30 caliber, I would get a 308. More ammo choices and unlimited platform.