What are the differences between a bone marrow biopsy and aspiration?

Nothing... They are literally one and the same... The aspiration is the procedure... And then the pathologist looks at the sample as a biopsy specimen.
Level of detail. Bone marrow aspiration is similar to looking at a blood smear where cells are viewed individually and stainned. Bone marrow biopsy is a tissue sample with bone and the cells around it and it is more informative than a simple aspiration and will be diagnostic more often.