A big difference between a FET and a BJT is the FET is voltage driven and the BJT is current driven.
Bipolar transistors are really voltage driven. When doing small signal analysis, the transconductance (voltage in current out) at room temperature is very reliably the collector current divided by 0.026. The base current is treated as a parasitic effect and the current gain (hfe or beta) is mostly irrelevant except in saturated switching applications.
Another issue with the BJT is that it will have a limit to how low the collector-emitter voltage will go (collector-emitter saturation voltage), typically around 0.6V. If you have a high current LED this means the transistor will dissipate a lot of power. If you have a 12V supply then your LED will get more like 11.4V.
Bipolar transistor collector-emitter saturation voltage is much lower than 0.6 volts. The reason it seems high is that bipolar transistors have higher transconductance and can sustain a higher current density than MOSFETs which results in a smaller die size and higher bulk resistances. (1) When the die sizes are equal, the saturation voltage is competitive. (2) Losses in saturation for both devices are rarely an issue.
The above points to the major advantage of bipolar transistors. For a given current rating, they can have a smaller die and that combined with easier fabrication makes them less expensive. However in practice these days, both transistor types are cost competitive so the price difference is rarely a factor.
(1) Higher transconductance and smaller dies also mean better high frequency performance.
(2) This is why bipolar integrated regulators have not been replaced by integrated CMOS devices. The CMOS output transistors are so large for the same dropout voltage that the chips are just too expensive to be competitive except when quiescent current must be as low as possible