The  Bumble Bee has a smaller face and yellow bands on belly.  Although its hair may be sparse in some places, it has hair all over.  The picture on the left is not high resolution,  but the hairs on its abdomen show clearly in the high resolution photograph.  The Carpenter Bee has a shiny,  hairless abdomen (what we'd call its back) and a larger head.  The male carpenter bee has a  white spot on its head.  Unless you can get them to sit still on a flower for long enough to get a good look, their habits may be the best way to distinguish the two.

Both bees are pollinators and valuable to the gardener and farmer.  Bumble bees usually nest in the ground, in abandoned mouse holes, but sometimes in unused bird houses.  Carpenter bees have the unpopular habit of drilling holes in wood (like the fascia board of your house!) to nest.  The best way to encourage them to nest elsewhere is to paint and caulk your wood trim.  Contrary to what you may have heard about termites, Carpenter Bees like natural cedar!

Neither the Bumble Bee nor the Carpenter Bee is aggressive unless the nest is threatened.  Male carpenter bees may seem aggressive but are not able to sting.  Actually, it's quite safe to observe them close up on a flower, as I've been doing.  If possible,  put up with the inconvenience of having a nest close to home, and just make it an intriguing lesson in nature.