Continental crust is less dense than oceanic crust. The main reason is that continental crust is richer in SiO₂ and Al₂O₃ and poorer in MgO, and FeO, and Cao. These later oxides are denser than silica and alumina.

If you took the same volume—the same amount—of oceanic crust as continental crust then the oceanic crust would weigh about 10% more on average.

Oceanic crust covers most of the ocean basins, about 260 million square kilometers (km). An average thickness of the oceanic crust is 7 km, so there are about 1,800 million cubic kilometers of oceanic crust. But continental crust only covers the 250 million square kilometers of the continents and offshore areas that have continental crust. Its thickness is 40 km on average. So that's about 10,000 million cubic km of continental crust. Oceanic crust is only 10-15% denser on average, which isn't enough to make up for the size of the continents near the surface!

But that's not all! At the edges of tectonic plates oceanic crust sinks into the Earth's mantle through a process called subduction. That means there are pieces of old oceanic crust that are deeper in the Earth. These old chunks of oceanic crust have sunk since plate tectonics started; there is more oceanic crust in the mantle than near the surface of the Earth. While it's true some continental crust can be subducted, the amount is much less than oceanic crust. If we include all of the oceanic crust that is deep in the Earth then all of the oceanic crust probably weighs more than the continental crust.

I think the important property here is not the weight ("heaviness") of the crusts but rather the density, i.e. the weight per volume of a material. The density is what determines how plates behave when meeting at convergent boundaries. Of the two, the oceanic crust is denser, around 2.9 g/cm³ (vs. ~2.7 gm/cm³). The difference is due to the compositions, with the continental crust containing lighter elements such as silicon, aluminum, and alkali metals in greater abundance.

Regarding weights though, the densities can be combined with the volume of each of the crusts to determine which is heavier. It turns out that the continental crust, although it covers far less of the Earth's surface area, is much thicker (25-70 km vs. 5-10 km). Thus, the continental crust, with roughly twice the volume of the oceanic crust (~70% of the total crust) and only a slightly lower density, will have a higher weight.

Oceanic crust has a higher density than continental crust. Evidence that this is true (proof), and can be observed when an oceanic plate collides with a continental plate. At a convergent boundary, where an oceanic crust is colliding with a continental crust, almost always the oceanic crust will subduct (go underneath the continental crust). This is because it is heavier.

The continental crust is thicker than the oceanic crust, read here "Continental crust is also less dense than oceanic crust, though it is considerably thicker; mostly 35 to 40 km versus the average oceanic thickness of around 7-10 km."

Continental crust is mostly granite, which has a density of 2500 kilograms per cubic meter. Continental crust is usually about 30-50 kilometers thick.

Ocean crust is mostly basalt, which has a density of 2800 kilograms per cubic meter. Ocean crust is usually about 5-10 kilometers thick.

Do the math, and you will find that a square kilometer of continental crust weighs more, despite being less dense than ocean crust. Being less dense, however, means that the continental crust floats on top of the mantle underneath it. Ocean crust, however, by being denser, is able to sink when it gets subducted.