For some applications, you could use either. FETs however are rather more ``modern'' and have a number of advantages: they have larger input impedance (drawing very little current), better temperature characteristics, better performance at high frequency, better linearity at low bias voltage. MOSFETs in particular are used almost exclusively in large-scale integrated circuits because they use much less power.
Bipolars tend to be quite robust and are less sensitive to static than FETs (especially MOSFETs).