To answer your question - the plates "weigh" about the same. They have to be. If one plate is heavier, it will sink and displace the underlying mantle so that it pushes the lighter plate upwards. Note that these things actually happen - the Earth is not in dynamic equilibrium and various tectonic and surface processes cause this disequilibrium and movement.

But there isn't anything inherently heavier about the oceanic plate compared to the continental plate, or vice versa, when taken as a whole. This is the concept of isostasy (with a well illustrated Wikipedia page).

And pay attention to what you're talking about the continental and oceanic plate or the continental and oceanic crust. The two are not the same. It is correct that the continental crust is about double (or more!) than the thickness of the oceanic crust, but the continental plate is not double the thickness of the oceanic plate. The plate includes the crust and the upper mantle - together called the "lithosphere".

In a cartoonish way, we tend to think of continents as the "soap scum" that floats on the rest of the earth. It is lighter than the rest of the material, and so whenever a continental and an oceanic plate meet, the oceanic plate generally subducts and the continent overrides.

In concrete terms, the density of the basalt that makes up most of the oceanic plates is 2.9 g/cm^3, whereas the granite that makes up most of the continents has a density of 2.7 g/cm^3.