Both fine cartridges... probably my three favorite cartridges are the .243, the .25-06 and the .270. But they really are different beasts. Both have a very flat trajectory and both will result in lots of game taken if you practice, know the round and use good bullets and put them where they need to go. The 243 is porbably the best dual purpose rifle - one that can be loaded with light bullets to make an excellent varmint rifle and loaded with heavy bullets to make a fine, though maybe not spectacular, larger game rifle. Many whitetails have been taken by the 243, 50 or so by our esteemed moderator, Art Eatman. The 243 has also been known for some pretty amazing accuracy. Many long range competitors use it though it seems to be falling out of favor as the 6.5mms gain ground over the 6mms. Fine for all sorts of varmints, deer, antelope and certainly coyote. The biggest game I would personally feel comfortable taking with the .243 would be mule deer at less than 200 yards.
The .270 can be loaded up with tough 150 gr bullets such as the X Bullet and has taken plenty of elk though many will argue it isn's suitable. A hard penetrating round, a tough bullet and proper shot placement will lead to a dead elk. Easy as that. And while I'm sure many elk have fallen to the .243, I just wouldn't even consider it unless I was starving. And again, while I'm sure moose has been taken with the .270, I would at least question in more in depth before using it for that. They say more moose are taken in scandanavia with the 6.5x55 which, while a fine cartridge, is inferior in pretty much every respect to the .270 so I guess it would at least be arguable for moose.
They will have similar trajectories though the Comparing a 100 Gr .243 to a 130 Gr .270 (which are about comparable), the 270 will be a little bit flatter shooting but the .270 will have considerably more recoil than the .243 if that is a factor. Both are relatively light my comparison to the big kickers but I can shoot either my 243s all day with no ill effects while it's a bit of a challenge, at least for me, to put 80 rounds through my fairly lightweight .270 and stay completely accurate.
If I could only rifle, it would be chambered in .270 or I might drop down to a .25-06. These are both capable of taking pretty much any game that is likely to be seen within 300 miles of me but the .243 is not. On the other hand, if you are looking to do some serious target shooting and varmint sniping along with some larger game up to the size of 250-ish pound whitetail, the .243 will do you just fine.