Assuming same price for both type and no growth (sites) in the near future, here are what I can think of
1) Private line will go through MUX'es (mostly physical layer equipment), and you can run any routing protocol you want, BGP, OSPF, EIGRP, IS-IS, or static
2) MPLS will require PE from the provider, where you will require running routing protocol like BGP, OSPF or static
3) some customer I'd worked with will run encryption and GRE over MPLS to protect their data but I hardly see customer will have such security requirement for private line
HTH,
jerry