what is the main problem with a solar system model of the atom?

Answer 1

The mechanical orbital model of the atom does not account for the energy radiated by the orbiting electron. Accelerating charges (rotational motion is accelerated motion) emits electromagnetic waves, which removes kinetic energy from the charge. The orbital model results in rapid collapse of the electron orbit.

Quantum mechanics eventually solved this, but still retained the possibility of planet-like orbital motion. It was later when the uncertainty principle was discovered, that prevents us knowing where the electron is and its velocity both to high precision. So if the electron is located to within atomic dimensions, it's velocity will be so uncertain, that it can only be considered to exist in a "cloud" around the nucleus.

Answer 2

One problem is that the electron will slow down as it circles the proton (conservation of angular momentum), so finally the electron and the proton will collide and the whole thing will vanish.

Niels Bohr, who thought up this idea, postulated that electrons are only allowed to occupy certain "orbits." These are paths that are a certain distance each from the proton. If that is so, then the electron is "not allowed" to fall into the proton. This was his solution to "the problem."