At the simplest level, you can think of protons and neutrons as each being made up of three quarks. As far as we can tell, quarks, like electrons, are point particles -- they don't have any size at all. The quarks in each are held together by the strong force, which acts a lot like rubber bands connecting them together. They constantly bounce around, sometimes closer, sometimes further, but maintaining an average radius of around 1 fm (one femtometer, or 10^-25 meters) for both the neutron and proton.
Unfortunately, that model fails outrageously if you look at neutrons/protons at all closely. This is because in quantum mechanics, the quarks don't occupy a particular point in space, but are "smeared out" over an area. Also, thanks to e=mc^2, the strong force (our rubber bands above) actually manifests itself as an ever-changing cloud of "virtual" gluons, quarks, and antiquarks surrounding the three "real" quarks. It's a mess.
You have to think very carefully about what you mean by the size of a proton/neutron. The only way we have to look at them is to try and shoot other particles at them at see what happens. And it turns out that what happens depends on the type of particle and its energy, in a complicated enough way, that 'size' isn't a very useful concept. Nuclear physicists use things called form factors and structure functions instead.
With that taken into account, the neutron and proton are nearly equal in size. This is because, at the tiny distances in the nucleus, the strong nuclear force is much stronger than any of the others. Under just the strong force, protons and neutrons are effectively the same particle; this is an example of something called "isospin symmetry."
Answered by: Ali Soleimani, B.S., Physics Grad Student, Madison, WI
Proton: 1.6726231 x 10-27 kg Neutron: 1.6746286 x 10-27 kg
Our result is that the neutron has a greater mass by about 2.0055 x 10-30 kg.
Answered by: Thomas Anderson, Physics Undergrad Student, UT, Knoxville
'One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike - and yet it is the most precious thing we have.'