Why neutrons are neutral?
Protons and neutrons are both baryons, subatomic particles composed of three quarks. Quarks are smaller fermionic particles that make up protons, neutrons, and other hadrons.

Electrons are leptons, fermions that are not composed of quarks.

Quarks have mass, spin, charge, and (confusingly) color charge. Color charge is a quantum property that quarks have, and all hadrons need to have a net zero color charge. 

The proton is composed of three quarks: two up quarks and one down quark. Each up quark has a spin of $+\frac{1}{2}$, a mass of $2.3~\text{MeV/c}^2$, and a charge of $+\frac{2}{3}$. Each down quark has a spin of $-\frac{1}{2}$, a mass of $4.8~\text{MeV/c}^2$, and a charge of $-\frac{1}{3}$. The total charge of the proton is $2(+\frac{2}{3})+(-\frac{1}{3})=+1$. The total spin of the proton is $2(+\frac{1}{2})+(-\frac{1}{2})=+\frac{1}{2}$. The masses of the quarks do not add up because most of the proton's mass is in its confinement energy.

We can do the same analysis for a neutron, which is composed of one up quark and two down quarks. The charge of the neutron is $(+\frac{2}{3})+2(-\frac{1}{3})=0$. The total spin of a neutron is $(\frac{1}{2})+2(-\frac{1}{2})=-\frac{1}{2}$. Again the masses do not add up. 

Protons and neutrons are different because the are composed of different quarks. However... enough physics. Now for the nuclear chemistry. 


  Interconversion of protons, neutrons, and electrons


The proton is a stable particle with a lifetime at least $2.1\times 10^{29}$ years.

However, protons can be converted into neutrons (and electron neutrinos) by electron capture:

$$\ce{p+ + e- -&gt; n +\nu}_e$$

This requires an investment of energy because the rest mass of the neutron is higher than the rest mass of a proton by $1.29~\text{MeV/c}^2$.

Free neutrons are not stable, undergoing beta decay to a proton, electron, and electron antineutrino with a halflife of about 10 minutes.

$$\ce{n -&gt; p+ + e- +\bar{\nu}}_e$$

Most importantly, and fortunately, most neutrons bound in nuclei are stable. The protons and neutrons in a nucleus can be treated quantum mechanically analogously to electrons from several atoms bound in a molecule. There are not a collection of independent particles that do not interact. The nucleons occupy quantum mechanical energy levels. Neutrons can only decay if the proton it would become can occupy a lower energy level, which is usually not possible except for those radioisotopes which tend to undergo beta decay, thanks to the Pauli Exclusion Principle! Thus, the bound neutron is stabilized.