To me, roam has a flavor of being unintentional about location but not necessarily unintentional about purpose. It's also more frequently used to describe the movements and activities of animals.
Roaming animals are very busy and intent on their animal priorities, but they don't necessarily care where they are doing them. So to me, roaming can include things like looking for food, sex, seasonal migration or satisfying curiosity. But I wouldn't get those kinds of flavoring from wander.
On the other hand, wander implies moving without intensity of purpose either in direction or activity.
Wander often implies slow speed, but not always. Whereas for me, roam would imply the typical speed of an animal grazing or hunting depending on the habits of their particular species. Roaming buffalo and roaming tigers don't imply the same speeds at all to me.
ghoppe pointed out some additional meanings of wander. Some additional meanings of roam includes moving outside of your normal cellular phone network. However, unlike ghoppe I think roam can also mean "to be sexually unfaithful."
Both wander and roam can be used to imply movement outside of expected or accepted boundaries or paths, but neither has to mean that, and that kind of meaning depends on context. In those contexts they are synonymous with the verb stray.
For example, unwelcome grouping could be identified as wandering or roaming or straying hands.