What determines the physical properties of elements, such as density and melting points? And is there any variances between isotopes of the same element?

The physical properties are all setup by the atomic structure. For instance, the number of protons and neutrons but also specifically their electrons and their placements (the number of outer shell electrons really makes a difference).

And yes, there are, usually slight, variances between isotopes of the same element. This is due to the change in neutrons from the base element, so that their overall structure is mostly the same, but their mass is different which plays a role in the physical properties.

https://www.ck12.org/flexi/physical-science/Chemical-Properties-and-Chemical-Reactions/what-determines-an-atom%27s-chemical-and-physical-properties/

https://www.geeksforgeeks.org/chemistry/why-isotopes-have-different-physical-properties/