So the big difference here between a rifle that is legal to hunt elephants with and a true elephant gun is a matter of the situation and circumstances in which the shot is encountered.
While it is true that a .375H&H is the minimum legal cartridge which is legal to sport hunt elephant it can not be relied upon to stop a charge at close range in thick cover.
Even the venerable .470NE is considered an unreliable stopper. All of the modern day big time elephant hunters that I've spoken to about this subject tend to agree that a reliable elephant stopping cartridge starts with a .50.
Most of the guys out there that are regularly relied upon to stop a big angry bull elephant at close range are using a minimum of a .500 NE a few shoot .577NE and there are a smattering of bolt guns in .500 Jeffery, .505 Gibbs ETC.
What you won't find are any pros shooting the .700 NE the .700 NE is a stupid human trick there are only very few of them built they are prohibitively heavy and here's the shocker they don't have very good performance on game and tend to lack penetration. While there are a very few .600 NE's around I know of only one pro who uses one. The .600 is also faulted for poor penetration.
Conventional wisdom stated that a reliable elephant gun should be a minimum of .400 cal firing a 400 gr bullet of solid patch construction @ a minimum of 4,000 Ftlbs of ME.
Modern bullet construction has increased those parameters somewhat.
My personal comfort level when I am hunting thick Jesse bush are the following. I like a fast handling weapon preferably a double or a bolt gun with either ghost rings or good shallow v express sights. IN a double I like it to be a minimum of .450 cal firing a 500 gr solid of excellent construction and in a bolt gun a minimum of .400 cal firing 400 gr solid at something over 2300 FPS. I like to be real near 5,000 Ftlbs ME or above.
If a guy were going to seriously get into elephant hunting my thoughts are that he's best be served by either a .500 NE double or in a bolt gun a .458 Lott, .450 Rigby or a .500 Jeffery. My preference being the .450 Rigby or the Jeffery. The more I play with the Lott the less I like it. I've killed two cape buffalo with them and played with them quite a bit. The .450 Rigby is just so superior a round that it would be my choice if I was to get into another .458 cal bolt gun.
If any of you guys are really interested in the process and procedures of hunting the African elephant there is a DVD out there called "Hunting the African Elephant" by Buzz Carlton. This is the most informative no BS instructional hunting film that has ever been made. There are multiple example on film of of .470 and .458 class rifles failing to stop an elephant where the exact same placement with a .50 cal or larger sporting round drops them like a ton of bricks. It is really pretty eye opening.
it can be found here.
http://www.cmsafaris.com/