Arboriculture involves the care of individual trees, while urban forestry deals with forests as systems (groups of trees) in a developed setting.
The science of arboriculture focuses on proper tree planting, pruning, fertilizing, water and other maintenance issues. It also focuses on overall individual tree health. Urban forestry is also a social science, as well as incorporating arboriculture, and includes landscape level management such as urban forest inventory, valuation, planning, policy, etc. The two fields certainly overlap, but the training and practice of both are unique.
A professional in the field of aboriculture is called an arborist and he or she may or may not have a college degree. An urban forester usually earns this title after at least the completion of a specific undergraduate degree.