While I love the .303 British and have taken many moose with it, the .308 Win is a superior design and offers better velocity, accuracy and bullet selection.
I look at it as a case of old tech vs. new tech.
At the same time, most rifles the .303 Brit are found in, cannot safely handle the pressures involved in really hot-loading the cartridge (P-14 is the exception), especially the rear locking Lee-Enfield design. The .308 is a very efficient design and it is generally agreed that it is an inherently accurate cartridge. It is chambered in many modern designs that can handle much higher pressures than the .303 Brit. The rimless design makes loading magazines much simpler as well, when compared to the Brit's rimmed design.
In practical field/hunting terms, however, I would not feel under-gunned toting a nicely sporterized Lee-Enfield. Shots out to 300 yards are no problem with the .303, although I'd prefer to get a little closer if possible. For the part of the moose, deer or caribou...the .303 will kill just as effectively and cleanly as the .30 caliber.