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Both ..
With source follower you may expect higher input impedance ..
As far as the output impedance is concerned, (1/gm is the output impedance of the FET) typical values for FETs are about 100 Ω whereas in bipolar transistors the equivalent value is 25 mV/Ic, about 10 Ω ..
That's all ..
If you consider AC or small-signal condition, Emitter Follower or Source Follower has not difference at all when buffering or mirroring the input signal to the output.
However, when you consider DC condition, source follower uses a MOS, so you can consider no diode drop between Gate and Source, hence Vout at Source is almost Vin at Gate. Because between Source and Gate is a Cgs capacitance, your Vout is better retained or buffered longer.
This examples why memory devices these days uses 1T or 4T MOS-Caps to store information in bits.
Source followers aren't good buffers due to the body effect .For the best case where an ideal current source is placed at the source of the MOSFET, the voltage gain will be :
Av=gm/(gm+gmb) i.e Av will always be smaller than unity. While for the case of Emitter follower, Av=1 for the case of ideal current source .
Also, source followers suffer from substrate coupling from surrounding circuits on the same substrate specially digital ones .
I would opt for a source follower over an emitter follower mainly because it has a much higher input impedance. Achieving unity gain using either source follower or emitter follower isnt possible unless you have an absolutely ideal current source at the source/emitter, which isnt usually the case. So I would rather go for a buffer with a higher input impedance which is essential to isolate the circuit at the input of the buffer, from the circuit at its output.
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