There is a clear and distinct difference between the Biomedical Sciences and Biomedical/Bioengineering. The Biomedical Sciences are concerned with the facts, theories and models describing biological and/or clinical phenomena, including subfields such as physiology, virology, biochemistry, molecular biology, or proteomics. Biomedical/Bioengineering applies traditional engineering techniques in subspecialties such as mechanical, materials, or electrical to solve biological or clinical problems. So the difference is as clear as, say between chemistry and chemical engineering, where one is theory and the other is practice.
The difference between Biomedical Engineering and Bioengineering is more subtle, and can have varying interpretations. Generally, Biomedical Engineering is the more widely encompassing field, including the development of engineering solutions to both biological ("Bio-") and clinical ("-medical") questions. Biomedical Engineering departments at universities typically have research in areas as diverse as tissue engineering, biomaterials, kinetic modelling, cell signaling, neural implants, and medical imaging instrumentation.
Bioengineering is generally recognized as a subset of Biomedical Engineering, and concerns itself with more of the scientific - primarily the biological questions, and less the translational ones. Kinetics, microfludics, physiological modelling and cell targeting are some standard Bioengineering areas; fields such tissue engineering, drug delivery and biomechanics are also considered Bioengineering because of their strong dependence on the basic science, but have more explicit translational/medical applications so they blur the line and add to the confusion.