Is midface suspension the same as a subperiosteal lift?

The extent of the facelift operation is tailored to each patient's needs and can range from:
A mini lift: a small skin excision from in-front of the ear that tightens the skin along the jaw line in younger patients, to 
A lower face and neck lift: an extension of the mini lift that brings the incision higher in front of the ear to the sideburn and further back behind the ear into the hairline to tighten the neck and reposition the deeper soft tissues of the face to efface neck bands, lift the jowls, and add volume back to the cheeks (a cheek lift), to
An midface and lower face and neck lift: a further extension of the incision higher in-front of the sideburn to allow tightening of the upper face — addressing the crows feet and midface.  
A sub-periosteal lift refers to mostly historical surgery where the facial tissues are lifted off the bone in order to elevate them. As our understanding of the details of facial aging has improved, the surgery has fallen out of favor because the surgical technique ignores the fact that the deeper tissues do not sag with aging, because it has a high risk of complications, and because the recovery from surgery is very prolonged and painful. 
See the link below for more info on facelifting.A detailed examination will help delineate the best surgical option. Consultation with a surgeon certified by the American Board of Plastic Surgery would be the next best step.

These are different procedures, done for different reasons, and have differing recovery periods. Mid face suspension involves lifting the mid-facial tissues and fat pad, and suturing them into a higher position. Subperiosteal face-lifts have fallen out of favor because of the excessive swelling and prolonged recovery time associated with this procedure.  Your plastic surgeon will discuss with you the different procedures that will improve your facial appearance.

Midfacial suspension can be achieved with  a variety of techniques including a subperiosteal midface lift, a high SMAS lift, or suture suspension midface lift.  Often times, I use volume (Sculptra, Voluma, or fat) to lift the cheeks without need for surgery.  The term subperiosteal refers to the approach to the cheek, lifting the lining of the maxillary bone; I rarely use this approach due to the benefits of alternative techniques these days.