I have owned, reloaded, and hunted with both 30 caliber rounds and have to say that they are both very effective and capable for almost all hunting applications.
The first comparison I can offer is felt recoil, the 30-06 is a little bit more than 308 in this area. So For new shooters still getting use to shooting with a scoped rifle 308 will probably be the best option.
Secondly I have to mention the cost factor. In today's market 30-06 is primarily sold for hunting purposes only, this is not the case with 308 win,there is a large variety of military surplus ammo cheep and plentiful. This cost advantage gives you the benifits of being able to shoot more ammo, so more practice is being done at the range.
The 308 cartridge operates at a lower barrel pressure so a typical barrel 308 can last 5000 rounds before accuracy decline is noticeable, where as a 30-06 is typically half that ammount.
308 win has a good balistic coefficient so it is accurate and stable bullet. If you reload for 308 or 30-06 the options of bullet types and powders avaliable are rediculous.
Either round is very good and will serve you well but I think 308 is the best option for almost any thing you can use it for.