Both flavors of support have certain team comps which will amplify their success. Tanky supports in general do have a more effortless feel of contributing to a win in a disorganized environment and are more generally easy picks across any meta. Right now I don't think I'd say that most of the squishy supports are outclassed, though.
I believe should be a more situational pick as he really isn't the most reliable lane partner. You usually would pick him more for the rest of your team comp and with a more self-reliant carry. His synergy with hard scaling fighters is really good. Personally, I don't think most of the tank supports give him as much issue as the heavier poke and sustain supports who will out trade his damage and force his carry back.
are ridiculously good at saving their teammates. Lulu particularly is notoriously impossible to outtrade when she is strong. Janna doesn't offer as much offensive pressure, but has huge disengage at all points of the game. Soraka is more vulnerable herself than any other squishy support, but is great at making a strong front line stronger and adds some global pressure.
and
are both generalists. The former more defensively-oriented. They can both flip more easily than the others between offering defensive or offensive pressure, though.
Frankly, if tanky supports are able to bully out the others in lane it's usually simply due to inferior positioning and strategy on the opposite side or combination with pressure from the jungle/other lanes (Not taking into account numerical imbalance, of course).