The tsunami is like the surge in that it raises ocean levels and moves inland. However, a storm surge also has waves coming in on top of it and very strong winds - so IMO it is more destructive for the same sea-level rise.
But it also affects a much smaller area. And there is plenty of warning for a hurricane, for those who choose to heed them.
The surge height of this in Mississippi was comparable to the tsunami in Sri Lanka and Thailand (but 1/3rd the height of the tsunami in Banda Aceh.)
While you have wind waves on top of hurricane surge, in tsunami surge you have it coming in and rapidly draining over and over again (remember the incredible force of the video of the tsunami DRAINING out to sea) surge just comes in, so in terms of physical force I suspect a 20 foot tsunami is worse than a 20 foot surge, marginally.
The key aspect as you note is warning, and the suddeness of the tsunami, in terms of human death tolls.
Good point with the waves and wind on top of surge. I dunno, I think we are going to see whole coastal areas completely wiped out when they get to them.