A differentiation is not by definition, duden.de uses it both ways:
- abnehmen, first meaning:
niemand nahm [den Hörer] ab (nahm das Telefongespräch entgegen)
- abheben, meaning 1.a
den Deckel, den Hörer abheben
Ich nehme nicht ab. - vs. - ich hebe nicht ab. (while phone is ringing)
Da nimmt keiner ab. - vs. Da hebt keiner ab. (while phone is ringing)
Works both ways. Just with your example "Nimmst Du mal bitte ab..." it is also implied to receive the call and talk.
Thus I would use
abheben (as well as rangehen(Duden, meaning c)) in context with a telephone more often that I started the call and the receiver needs to react and pick up the phone to receive my call.
The other way round
abnehmen I would use more often when someone else is calling and I should pick up the phone. It is similiar to
annehmen which means I take the call. This refers to the times when a switchboard operator(wiki) was necessary to establish the call: the operator would ask me "if I take the call?" and I still could say "no, I don't take the call".