Both are great cartridges.
Ive had 4 25-06's in the last 10 years and 2 243's.
I've shot everything from squirrels to elk with my 25-06 and everything in between you can hunt in ORegon. Made 1 shot kills on deer and coyotes out to 500-600 yards regularly with 115g nbt's at 3200fps.
I think if you want 1 caliber to do both great, take the 25-06 and dont look back. There accurate, flat shooting, little recoil, and hit a lot harder then most think. 90% of the animals I've shot with my 25's just drop DRT to shoulder shots.
I think if you just want a dedicated coyote rifle though, you cant beat the 243. With 55's at 3900-4000 they shoot amazingly flat and knock the snot out of coyotes out to 400 yards or so.
The 243 is an ok deer cartridge out to 300-400 yards I'll give it that, but not much more.
It does also have the advantage of the 105amax and those type bullets to get you out to 1K relatively easy. With the 25-06 you only got the 115g vld and your burning 10-15g more powder.
25-06 was my first rifle I started getting into LR shooting with. Its been my favorite cartridge for the last 10 years and I always said I'd never be without one. However I sold my rem 700 24" shilen barreled 25-06 last night. It was 1 sweet shooter, 100g nbt's at 3500 and 115g nbt's at 3250 both into .5 MOA.
I like to have a rifle for a specific purpose. 243 IMO is one of the best coyote calibers out there hands down. My stevens 243 shoots .5-.75" with reloads and a trigger job. I find myself taking that over my 25-06 90% of the time now. ALso for my long range deer hunting, my 7mm RM is just a lot better choice for me and my type of hunting, so the 25-06 had to go.
However, if I was limited to only having 1 rifle caliber, it would probly be a 25-06. With the right bullets, it will take squirrels to elk out to 500 yards in the right hands.