IMO the only advantage is that you have less rotational mass and the wheel is easier to get it rolling, besides that probably the aero gain part is minimal compared with a 700c wheel.
We got TT track funny bikes back in the day with 650 wheels in the back, 58 chainwheel to compensate for the wheel, the bikes were pretty easy to get them rolling but once in the race personally I did not notice big differences in relation to a regular aero track bike, same stuff.
650 is a good creation of short people so they are able to get some seatpost showing and a descent fit, besides that the % of improvement unless we are talking into super high performance atletes, probably the regular john doe wont even notice the difference.