The short answer is try the 165-168gr loads.
A big consideration in what bullet weight to pick is the twist rate.
Most 30-06 barrels have a 1:10" twist, which usually work best with 165-168gr bullets.
BUT, it's not really the weight that matters, it's the bullet length. So, if you use something like all copper Barnes bullets then you drop down one weight size and use the 150gr. Copper bullets are longer than jacketted lead bullets.
You should try many loads. Start off with the lower cost 150 and 165gr soft points.
Also try Federal Premium Ballistic Tips. They tend to shoot well in many guns. The 150gr Ballistic Tips shoot very well in my 308, but most other 150gr loads don't shoot as well as the 165-168gr loads. The 150gr Ballistic Tips shot well in my 30-06 when I had it.
The Federal Gold Metal 168gr BTHP and 168gr BTHP Black Hills Match are generally very accurate factory loads. You may also want to try the Gold Metal 175gr BTHP.
Besides the 15 grains is there any real differance between the like bullets in the same weight?
A few bullet designs will have slightly tougher jackets in the heavier bullets. Tougher jackets and slightly lower velocities would mean more penetration and less expansion. That doesn't matter when you're killing paper though.
Heavier bullets also tend to recoil more even though they go a little slower.
When you start shooting at very long ranges then you'll also see that heavier bullets can remain stable and resist wind longer than lighter bullets resulting in better accuracy.
There's a lot to learn about shooting, but you start by buying some cheap ammo and going to the range to experiment and get some experience.