Intuitively, my interpretation of manboobs would be the more "realistic", usually oddly-shaped, gynecomastia type, while male_with_breasts would be more like a man with cartoonish breasts bolted on through some magical or medical means (to the extent that "cartoonish" means anything when it's all 2D). To illustrate, I see a bunch of ikemeru19 art in those tags; the delivery boy's breasts would be manboobs, while if he somehow ended up with that artist's brown girl's huge tits on his chest, I'd call him a male_with_breasts.
But that's probably not a practical distinction to enforce, so it should probably all just go under one tag. While the male_with_breasts does have the advantage of being less slangy, it looks like the vast majority of material is more in line with the mental image manboobs gives, and so that's probably more appropriate.
Then again, that's just my personal intuition.