A graft or a flap can be made up of the exact same tissue type-- that is, it can be skin, fat, tendon, bone, nerve, etc. The key difference is that a flap has its own blood supply, and that a graft does not, and therefore requires a good vascular bed for it to survive. We have often heard of the terms skin graft, or bone graft. These tissues are harvested from their specific locations which can afford to give up a part of their architecture. The place that the area has "donated" its tissue is called the donor site. Once the donor site has been harvested, the skin or the bone can be placed in the new area for the body to accept this tissue and eventually grow into it with new blood vessels and adjacent cells. This is why the bed, the accepting area for the graft, is so important to be prepared well. The graft can only survive well when it can be nourished by a good blood supply.
A flap has the additional advantage that it has its own intrinsic blood supply, allowing it to be transferred or transplanted to another area of the body with much less reliance upon the surrounding tissue bed. This does not mean that the surrounding bed should not be well prepared, ie debrided. It simply means that the accepting area will unlikely lead to graft survival and that designing tissue with its own blood supply is the only way for that tissue type to actually survive in the area. Because a flap has its own blood supply, the surgical technique to harvest and to move this tissue is much more complex. The surgeon has to dissect not only the block of tissue to be moved, but also the blood vessels (artery,vein) that feed the tissue block. This makes the flap an extremely powerful tool in reconstructive surgery. It is also a tool that requires significantly more surgical technique, planning, and time to make it successful.
It is understandable why there might be confusion between these two concepts because of their relationship to each other. However, the complexity and time factor between these two concepts is so significantly different that we should not pass over this as simple semantics.