They're both excellent knives with excellent locks and steels which impress most knifeknuts. I have a full-sized Manix and like how it locks into my hand in a variety of grips. Its wide blade allows for extra mass to help with slicing and pushcutting. Its lack of thumbstuds allows for the choice of most any sharpening angle and a lack of interference while cutting. The 710 will more likely have less (if any) spine-to-blade movement when locked (it's not a safety concern, but it does feel annoying at times). The 710 is designed to be flicked open and closed. Its blade has the extra slicing power of a recurve (compared to non-recurved blades with similar shape and mass). The 710 also takes up less pocket space (compared to the full-sized Manix, at least). The Axis Lock is less likely to be fouled by dirt or lint than the Mini-Manix, but lint-related lock failures on Spyderco lockbacks are very rare and easy to see coming (and easy to fix with a cotton swab/Q-Tip).
Over either the 710 or Mini-Manix, I'd suggest the Delica, Endura, or Calypso III from Spyderco. They'll give the same, or better, cutting power and weigh less and cost less (especially the Delica).