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.
Long range shooting is a process that ends with a result. Once you start to focus on the result (where the shot goes, how big the group is, what your buck will score, what your match score is, what place you are in...) then you loose the capacity to focus on the process.