Yes, everything changes when you start adding quarks, which do have spin. What essentially makes a proton different from a neutron is the charge and the spin, or more precisely the quarks that make up each. Since Quarks are the fundamental components of both protons and neutrons their spins can be added. The quarks involved in both are the flavors up and down. Up quarks have a spin of 1/2 up and a fractional charge of +2/3, while down quarks have a spin 1/2 down and a fractional charge of -1/3. A proton is formed from three quarks two up and one down, similarly a neutron is formed of two down and one up. A deuteron or for that matter any nucleon can have spin because you are just adding quarks. Also, quarks are held together by the strong force which has associated with gluon particles. If you want to learn more look up Quantum Chromodynamics or wikipedia quarks.
http://en.wikipedia.org/wiki/Image:Quark_structure_proton.svg