This is going to take a bit, so be prepared. Also, some of what I say will not be exactly correct, in order to get the larger point across, but bear with me.
First, the Rutherford model really did not speak to the question of emission lines. It simply noted that, on the basis of Rutherford's scattering experiments, all of the protons in an atom had to be concentrated in a dense clump. Before this, the fact that protons repel each other suggested that an atom was made of separate protons (which repelled each other) and electrons (which also repelled each other), while the attraction between the protons and electrons kept the whole assembly from flying apart. This was called the Plum Pudding Model.
In this model if you try to fire an $\alpha$ particle through something like gold foil, the $\alpha$ particle ought to hit a large number of small obstacles, and this would produce a specific scattering effect. When Rutherford actually did this, he discovered that this isn't so - the experiment behaved as if there were a small number of very large obstacles. Rutherford concluded that, although it made no immediate sense, the protons were somehow clumped together.
Shortly thereafter, Niels Bohr pointed out that this would make sense if the atom's electrons, being attracted to the clump (which shortly became known as the nucleus) of protons, could be thought of as orbiting the nucleus, just as planets orbit a star. This became the Rutherford-Bohr model. At it happens, it made all kinds of sense out of the known behavior of atoms, and was quickly adopted. It is still taught in grade schools and high schools, even though it has been overshadowed by the more accurate and powerful ideas of quantum mechanics. You can think of it as being like the difference between Newtonian and relativistic physics. For some cases, Newtonian physics works just fine. For some analyses the Rutherford-Bohr model works just fine, too.
Thinking of an atom as a little solar system, you can think about the energy of the planets/electrons. Let's take Mercury, for instance. If you put a really big rocket on Mercury, you could speed it up and cause it to move to a larger orbit. The increase in speed means an increase in energy, so you can say that larger orbits have more energy than smaller orbits.
This leads to the idea that, when an excited electron emits light, it loses energy and falls to a lower (smaller) orbit. With me so far? Now a problem comes up. If you look at the emission spectrum in anna v's answer, you may notice something: there are only a few sharp lines visible. This means that, in an excited gas which is emitting light, billions of electrons are all emitting exactly the same amount of energy when they "de-excite". Think about what this means in the Rutherford-Bohr model.
First: Each hydrogen has one proton in its nucleus, and all protons are identical, so for every hydrogen "solar system" the central sun is exactly the same - check.
Second: Each hydrogen has one electron as a planet, and all electrons are identical, so for every hydrogen "solar system" the planet is exactly the same - check.
Third: Since each "de-excited" planet gives off one of a small number of specific energies, the orbits found in each hydrogen "solar system" must be the same - che - wait, what?
Why in the world would this be true? In a macroscopic solar system, you can get any orbit you want by tweaking the speed of the moving body. So why would electron orbits be restricted to a small number of possibilities?
Well, that's what quantum theory is all about. The energy that a bound electron can have is restricted to integral multiples of some base energy - it is quantized.
And that is why the Rutherford-Bohr model does not explain emission lines. In the case of the hydrogen atom, an electron can switch between different energy levels as it emits light, and the relationships between the possible energy states of the electron produce a well-defined relationship between the different emission spectra. The Balmer and Lyman series are examples of different classes of orbit-switching.
