The reason why any of these word would be used in place of “breast” can be either due simply to personal choice, or there could be a level of immaturity about it, much like how many people avoid the word “moist”.
That aside, it could also be because they are using these words for their more exact/traditional definition.
“Breast” is a rather broad term that can refer to pretty much everything in that general region in the body. Thus we have words like breastplate, which almost invariably (yes, I’m sure there were exceptions, but there were just that: exceptions) was meant for men, therefore having no connotation of a woman’s breasts (notice, it really only takes on this meaning when made plural).
“Bosom” is often used as a replacement, but it is actually more accurately used when referring to this region when bringing someone/thing into a welcoming, or nurturing embrace. Obviously, the image of a mother holding her child to her bosom has, over time, brought with it the connotation of a woman’s breasts, but this is not, necessarily, the original intent.
“Bust” actually extends beyond this region to include the head, originally used in sculpture. I believe, although I could be wrong, this is the term often utilized in tailoring, and thus has received a connotation for the breasts, for obvious reasons.
Whereas all the others generally refer to the exterior, “chest” technically refers to the internal portion of this region, but it has also taken on this same connotation as all others mentioned above, simply because it is that general region.