Even keeping things classical, linking radio telescopes in "very-long-baseline interferometry" requires a bit more than just the normal internet (it requires synchronized clocks). But as you suggested, all this is doable classically.

However, all you get with linking telescopes classically is a "virtual" telescope with a better resolution (the angular resolution is roughly proportional to 1/radius of telescope opening, and if your linked telescopes are on different continents you get "virtual" radius as big as that of Earth).

This scaling of the resolution of 1/radius is what gives you the "diffraction limit". However, if we have sensors that are entangled with each other, we can surpass that limit. Googling "quantum entanglement diffraction limit" would give in-depth information.

If you wish to transmit quantum information then you need a quantum link (or entanglement plus a classical link -- google quantum teleportation). One use case is to do distributed quantum computing -- if the quantum computers are not colocated then they will need to be connected with some sort of quantum network