Neutron mass: 1.008664 u
Proton mass: 1.007276 u
Why the discrepancy?
On a related note, how does one go about measuring the mass of a neutron or proton, anyway?
Masses and coupling between quarks are free parameters in the standard model, so there is not real explanation to that fact.
About the measurment: you can have a look at this wikipedia article about Penning traps which are devices used for precision measurements for nucleus. Through the cyclotron frequency (Larmor factor) we can obtain the mass of the particle.
Edit: "A neutron is a proton + an electron" is a common answer to this question, but it is a totally invalid reasoning.
Both protons and neutrons are made of three quarks. The mass of the quarks is not known with enough precision, and even more important (and that's a why for the masses of the quarks), the interaction between them is responsible for the mass value to a much larger extend.
A proton is made of two up quarks and a down quark, whereas a neutron is made of two down quarks and an up quark. The quark masses contribute very little of the actual mass of the proton and neutron, which mostly arises from energy associated with the strong interactions among the quarks. Still, they do contribute a small fraction, and the down quark is slightly heavier than the up quark, which approximately accounts for the difference. (The masses of the up and down quarks, because they're so small, are not extremely well-measured, and detailed calculations of what the proton and neutron masses would be for given quark masses are difficult. So it's hard to be quantitative about the answer in a precise way.)
You can read a bit about the state of the art in calculations of the proton mass here: http://news.sciencemag.org/sciencenow/2008/11/21-02.html