I am new to this forum, but I am not new to hunting.  I have never shot the 708 but I have experience with the 7mag and many other 30 cal rifles.  In my humble opinion the 708 is not a good beyond 400 yard big game rifle.  It is just not going all that fast and if the shot placement is off just a little you are not going to get lots of penetration.  I have heard accuracy is great and most shoot the round well due to light recoil which is important.  I like the 7mag but it is a jump up in recoil.  I would feel more comfortable with a 500 yard shot with a 7mag.  All of my longer shots though have been with a 300WSM or 300 Weatherby.  All my 308 kills have been close.  I would also only hunt with premium bullets.  Good luck.  Oh and sorry I have not shot a T3 in anything more than a 25-06.  My guess is the 7mag kicks pretty good. 

If you get a 7mm Rem Mag in a T3 Lite it is going to be a pretty extreme recoil experience. I have the T3 Lite in 7mm-08, and due to how light the rifle really is, it kicks way above what I consider normal for caliber with factory loaded ammo. I have another 7mm-08 that's just a couple pounds heavier but is absolutely a kitten compared to the T3 Lite. The recoil pad as manufactured leaves a lot to be desired in the arena of recoil mitigation. All that being said the T3 in 7mm-08 is still not a bruiser, by any stretch of the imagination, just more recoil than I've grown accustomed to in that caliber. If you're looking at a Tikka with a wood or laminate stock then you can probably disregard all of that.

As far as the caliber goes, 7mm-08 will kill anything Elk-size and smaller out to 300 yards/meters, but beyond that you really have to know what your ammo is capable of. One of the drawbacks to 7mm-08 as a cartridge is the lack of variety in available factory ammo, (would you like 120, 139/140, or 150 grain?). If you don't reload, in addition to more energy for longer distance shots, you're just going to get more versatility out of the 7mm Rem Mag based on ammo availability (at least here in the States). I am a certified 7mm-08 nutjob though, so take it for what it's worth, internet rambling.

As far as the caliber goes, 7mm-08 will kill anything Elk-size and smaller out to 300 yards. I am a certified 7mm-08 nutjob though, so take it for what it's worth.

Nothing wrong with that. ;D

All I build anymore, for hunting, are .284WIN and 7MM-08's. 8)

Have you considered 300 WSM? Nearly identical in exterior ballistics to the 7mm Rem Mag in a short action. I'm a fan of 7mm-08 but its real appeal is at loner ranges on mule or whitetail deer.

I have  a boyds thumbhole on my 300 WSM, with a limb saver recoil pad, can shoot it all day. The thumhole makes it a pleasure to shoot.

I am a huge fan of 300 WSM.  In fact that will most likely be my next T3 purchase.  Just as soon as I find the right price.  I have had great luck with the 300 WSM taking down big game from small WT deer to large elk and oryx at distances from 100 yards to 400 yards.  168 grain Berger VLDs have been the most used bullet but I am researching the 190 VLD currently.  Good luck with your choice.  I am sure your 7mm will be just fine.