1: Some smiths will not chamber the 338 Lapua on the Remington 700 due to the very large base which if you dont want a 700 it is no big deal. If you do want to use a 700 for a project, the 338 RUM is one of the next best things.
2: 338 Lapua brass is phenominal brass. They can take a pounding and still hold a primer. They last along time and can be reloaded many more times than RUM brass. The quality and concistency of the Lapua brass is much better.
3: Both are very accurate (inherently) and both produce good muzzle velocities. However, there are a number of improved versions of the 338 Lapua that will blow the RUM out of the water. Stock for stock, they are pretty close to eachother all being equal.
4: Lapua brass is more expensive and harder to come by. For some it is worth it and for others it isnt.
These are only a few points to concider. Some time ago I was faced with choosing between what caliber I wanted and the smith I wanted to use. I really wanted a 700 action and the most important thing was my smith choice. I also wanted a 30-338 LM improved. The smith would not build the LM on the 700 bolt. I would have rather used that particular smith than use another and get my LM. I settled for the 300 RUM instead. One of these days though, I will have a 338 LM based cartridge chambered rifle in my safe.