How do leptons differ from one another? All the leptons that I have come across have their native (matter) charges of -1 relative to the charge of an electron. Since they are all fundamental and are not made up of quarks, you can't differentiate them based on their composition. Yet a muon is still nearly 200 times heavier than an electron. Does that mean that one is more fundamental that the other?
 A: The elementary particle table in the Standard Model distinguishes between three generations of quarks, leptons and bosons. 
The electron is in the first generation and the muon, a heavy electron, is in the second generation. An even heavier electron, the tau, is in the third generation. 
In the SM there is no good reason as to why there is three generations. The natural answer is either one or more. There’s a possible answer in String Theory: the higher generations are seen as resonances of the lower generations. 
A: Someone with a deeper insight into particle physics could possibly give you a better answer, but both are elemental particles. This is the same for the three generations of quarks. There are three sets of two quarks which have the same electrical charge. The only way then to differentiate them (except by their mass) is by their flavour. However the flavour, while of course being essential to solve the corresponding equations, is simply a note for the kind of particle, so different flavours for electron, muon, tauon.
A: The standard model can be derived by constructing the most general solution of quantum field theory (a combination of quantum theory and relativity) restrained by the fundamental symmetries of the universe. From such a treatment you can derive the existence of all the particles in the standard model. Some of their qualities are given by the model, and some (such as the lepton masses) are left as free parameters and have to be measured to complete the model.
All these particles are different. However, some qualities are shared. A number of particles have half-integer spin (and also charge -1). We can gather these into a group called the leptons. They still differ in other aspects (such as mass), but share this spin quality. Their difference is a fundamental consequence of the theory - if you plug in the electron in the muons place in the theory, your predictions will come out wrong.
Where does the difference come from? In the standard model, the difference between fundamental particles is simply a consequence of the symmetries of the universe. It may turn out one day that there are particles which are more fundamental than these, but until then, this is the best explanation we have.
Also note that differentiating them by their mass is completely fine, and often practical.
