From what I have been told the difference is barely noticable, however in some situations you will want to use one over the other. 카톡 메시지 보냈어 <- 부치다 would be strange/not work 어제 너에게 편지 보냈어 <- Even with actual letters you would most likely use 보내다 소포 부쳤어 <- almost only with parcels you will want to use 부치다 Short disclaimer, I'm not a native and this is just what Korean friends told me, it might not be 100% correct but I hope it helped from another learner's perspective!

It's really difficult to explain. That's why both of them are almost the same. Usually, we can use both of them. It's one of the difficult "Korean nuance", I think.