I always think this one is simple, the first thing is that both will do the job. The second thing is that, at all reasonable stalking ranges, you can down load the 308 to behave like (actually better than) a 243 with 110 grain bullets but you can't ever load your 243 up to 308 levels with a 150 or 165 grain bullet. With each 308 you buy you get a free 243.
I've never shot a deer and spent time thinking that I wished that I'd made a smaller hole in it but I have spent time after a not so good shot thinking that it was good I'd made as big a hole as possible.
Last season I had two BIG sika stags in the scope, I'm pretty sure both would have made a medal. I didn't shoot either of them as they would have been too heavy for me to extract and I mostly shoot for meat rather than a trophy. Neither presented broadside on shots or came closer than maybe 150 yards. At that distance a neck shot was out and it would have been necessary to shoot through the point of the shoulder on both of them. Had I of taken either shot I'd have been much happier with the 308 and 150 or 165 grain bullets than a 243 with a lighter bullet. Should you ever find yourself in the same position my money says you'll never regret having the bigger rifle with you.