I'm one that actually believes there is a difference. Both will kill game with ease, and well, but the .30-06 is more versatile, and a better case due to the longer neck- it was also made before fitting in detacheable mags was a problem, and guns were built around cartridges not vice versa. Those who think 200 fps mean crud all need to give their head a shake! We go to great lengths, AI'ing chambers, tweaking up hot handloads, Hornady 'Light Mag' ammunition, to gain those precious rises in velocity. If you handload , .30-06 hands down. I don't hold a bias here despite what it sounds like, my favourite small bore hunting rifle was a .308 for a long time. It works, but the .30-06 does work better, especially when the fellow's talking elk.
I'm stunned to see so many people, so readily recommending .308 for a guy who wants to include elk on his hunt list. How many have actually hunted elk that are telling him to go for it? I think some internet guru'ing is going down... I would consider .30-06 and a 200gr bullet (180's for Barnes bullets) the absolute minimum for elk.
As for cougar, most around our acreage in the Rocky Mtn foothills use .22 centerfires or .243's. You will never, ever get up on a cougar without hounds, the dogs tree the cat, and that's that a .224 takes them out of the tree. It sounds like the OP is new to hunting, let's not tell a new hunter to take a marginal caliber out and chase elk!
As for the Remington rifle, pick up a Winchester, and a Ruger off the racks beside them. Cycle the actions, look at the bolt handles and extractors, fit and finish, the safety ergonomics and function (we'll leave that whole other Rem 700 issue aside)... and I'd be absolutely stunned if you walked out with the Remington. Your money goes a lot farther in other brands these days.