Specifically, Bohrs atom model gives the radius of an atom as 0.53Å, while a quantum description offers a similar value. What is the difference between these?
The Bohr model is a comparatively simple model, whereas the quantum description is part of the huge theory of quantum mechanics, so to list all the differences including implications would need to explain quantum mechanics itself.
But I think quite a good starting point would be to say, the Bohr model has electrons travel on certain, specific, classical trajectories, picked out by certain rules. The quantum model does not have electron trajectories. Instead electrons are quite different, they are spread out a bit like clouds of gas, in areas called atomic orbitals. I would recommend reading about atomic orbitals, the Wikipedia page is quite interesting. You will see very quickly they are quite different from the Bohr model.