The bumble bee, the more "well known" of the two, has a hairy abdomen which is very visible in this photo.... Bumble bees generally nest in the ground in abandoned rodent nests, where a queen bumble bee creates a small colony of workers over the course of the summer. Reproductive males and females are raised in the colonies at the end of summer, and it is only the fertilized young queens that survive the winter to start new colonies the next year.
The carpenter bee, on the other hand, nests in unpainted or soft wood and has a generally hairless abdomen, which is easy to see in this photo. Carpenter bees eat out straight holes into wood, then create a few cells all in a line that they pack with pollen and nectar before laying 1 egg in each cell and capping it. Carpenter bees are often considered pests because of these holes that they create in the trim around porches, doors, railings, etc., but they also nest in dead trees and other wood.
Male carpenter bees, which do not have stingers (egglayers), are particularly noticeable and frightening at certain times of the year, as they zoom back and forth protecting their territory, waiting for females to come along so they can mate. Without stingers, their buzz is much worse than their "bite", but they certainly can be startling and rather disconcerting.
Both carpenter bees and bumble bees are important native pollinators. The photo below of a carpenter bee shows pollen dusting its head and body, pollen that will go along with the bee to the next flower where it can cross pollinate that flower and help create the next generation of plants, as well as the next generation of bees.
With honeybees having so much trouble these days and their populations seemingly in steep decline, native pollinators such as bumble bees, carpenter bees, syrphid flies, butterflies and so forth are all the more important. In some ways, I feel like my asters gathered all of these important insects together to reassure me that diversity and flexibility is the name of the game, at least in nature.