The .25-06 would have amazing potential if there were a large selection of heavy .25 cal bullets out there, but there aren't. The .243 has a pretty distinct advantage as far as being flexible.
.25-06 appropriate bullets range from 75-to 122 grains, and the corresponding construction of those bullets are appropriate for everything from varmints to elk.
The only place .244" holds an advantage is the higher BC of the match type bullets. That said, the much lower velocity of a .244" 115 gr. Berger VLD from a .243 (2,900 FPS max) causes it to have a worse trajectory than a 115 gr. .257" VLD out of a .25-06 (3,200 FPS max). Energy retention of the .243 will be better, percentage-wise, but they don't meet on the graph until the 1,000 yard mark, and at that range, the .25-06 will have about 30" less bullet drop when zeroed at 200 yards. With these max 115 gr. VLD loads, the .25-06 also boasts an extra 30 yards of MPBR.
Yes, the .243 is more efficient and a
little lighter recoiling in the same gun (often, though, the SA .243 rifle is a tad lighter). Performance-wise, though, it is simply outclassed by the larger .25-06.
The .243 will kill any deer out there and I know people personally that hunt elk exclusively with a .243.
For deer, yes, six of one, a half-dozen of the other. For elk, though, I won't go lighter than my .25-06. I have a 6mm rem, which edges out the .243, but I don't consider it an elk rifle.
I am building another 6mm rifle, but one that can really take advantage of the high BC .244" bullets. A 29" barreled 6mm-06 AI varmint rifle on a 700 action. This rifle should be fully capable of pushing those 115 gr. .545 BC Bergers to 3,400 FPS. But this is going to be a 13 pound varmint rifle, not a sporter.
The .25-06 has remained popular because of it's combination of light recoil, flat trajectory and excellent versatility. The .243 has similar attributes, but is simply a less powerful round, thus a little more limited on maximum range and game animal size.