The basic difference between the two devices, is that Bipolar Junction Transistors BJTs are current operated devices and FETs are voltage operated devices.
The main advantages of BJTs are that they have faster switching speeds and higher operating frequencies. ( BJTs historically have had greater power handling capabilities than FETs, but the power handling ability of MOSFETs has improved since their first introduction, particularly with the development of devices called "VMOSFETs").
The disadvantages are;
Virtually no current flows in the gate of a FETs (IGFETS have insulated gates), this give them the following benefits;
Disadvantages include slower switching speeds and lower frequency limits compared to BJTs. One particularly important aspect with MOSFETs that requires caution, is that static electricity can destroy or damage the thin insulation between the gate and substrate. Therefore handling precautions are required to eliminate static electricity.
(* When either BJTs or FETs are used to control large currents, then of course heat is generated. However in many applications in digital electronic circuits, transistors are just used in latching elements. Even in this application BJTs constantly require current flowing through them, while FETs do not. Therefore the difference in energy consumed, (and heat generated) in these circuits, is significant )