Until recently I would have said that the biggest difference is more in the quality of the brass/ammo. Ballistically they are very similar, but the crap that was available for brass for the RUMs I believe was where the Lapua gained a name for accuracy from the beginning, as the guys at Lapua understood good brass, good ammo makes for accurate rifles.
I have a 338 Edge which is 300RUM necked up as well as 3 338 Lapuas, ranging from a production Sako to a TRG , a full blown custom on a Nesika action and soon a HS Precision HTR.
The Lapuas I found shot far more accurately until last year when Nosler brought out what amounts to Match quality brass for the 300RUM. My 338 Edge which is based on a 700 SPS SS action with no modification other than a full true up job on the action, a RockCreek barrel in an A5 stock will hold its own just fine against the Lapua now.
All are capable of 2900 fps using 300 MKs
I love my Lapua's but if money were a deciding factor the 338 Edge would be the way I would go.
Actions are readily available, brass is less expensive, dies are more expensive though, but in the grand scheme the total package is less costly.
KK