Appearance
The simplest way to distinguish a carpenter bee from a bumble bee is to look at it closely. While both are roughly an inch an length and have heads, thoraxes, and wings with the same coloration, carpenter bees can be distinguished by their slick black abdomens. In contrast, bumble bees have furry abdomens with black and yellow stripes. In addition, male carpenter bees have yellow faces, while female carpenter bees and all bumble bees have black faces.
Hives
If you manage to follow a bee to its hive, this can also be a good indicator of what species it is. While bumble bees tend to live underground or in loose, fibrous material like leaf-meal, carpenter bees carve their homes out of wood. This can either be directly into a tree or into the wood paneling of a human structure, such as a barn or house. The carpenter bee drills a perfect hole into its habitat, leaving a smooth surface that almost appears man-made.
Aggressive Males
Female bumble bees and carpenter bees are generally non-aggressive and will only attack if their nest is threatened. Male carpenter bees, however, are unique amongst the two species for a strong territorial mentality. It will fly in the face and buzz immediately overhead of a person intruding on its turf. This behavior is purely threatening in nature, however, as the male carpenter bee does not have a stinger. It is simply warning you to go away. This behavior is not seen among male bumble bees and is a good identifier.
Honey
Of the American bees, the bumble bee is notably the smallest producer of honey. Although it still makes honey, the quantities are in such a small yield that farming them would be unproductive. In contrast, honey plays a much larger role in the lives of carpenter bees, which use it as a primary food for their young.
Size
While carpenter bees are usually an inch in length, bumble bees vary in size depending on the time of year. In the early spring as a new hive is forming, worker bees emerge much smaller than their full grown peers. These undernourished specimens often die within a few weeks, but not until after they've gathered pollen for the next generation, which is noticeably larger. This cycle persists until the late summer, at which point full grown bumble bees can be seen in flight.