1) According to the Rutherford model, the electrons revolve around the nucleus in closed paths similar to the way planets revolve around the sun. However, this model has some serious difficulties.
According to the Classical EM theory, any accelerating charge should emit radiant energy in form of electromagnetic radiation.
Since the electron in an orbit is continuously changing direction (hence accelerating), it should emit energy and this energy comes at the expense of the electronic energy, Hence, as the electron radiates energy, it loses it and orbits finally get smaller and smaller and the electron collapses into the nucleus leading to collapse of the atom.
This shows that Rutherford model is very inconsistent with the ideas of classical electromagnetism.
2) Atomic spectra of various elements consisting of discrete spectral lines was a well known experimental fact by the late nineteenth century.
Classical Physics such as the Thomson or the Rutherford atom is unable to explain the origin of spectral lines in the atomic spectra.
According the the Rutherford's solar system like model discussed above, the electrons should revolve around the nucleus in closed paths. However, predictions of classical EM theory suggest the electron to emit energy in a continuous manner as it spirals it's way into the nucleus. Therefore, such a spectra should be continuous and not discrete.
It was later understood that discrete spectral lines was a direct evidence of quantized atomic energy levels and could only be explained by the rules of quantum theory.