Patent fingerprints are made by a liquid or powder that sticks to the finger and then transfers to a surface, leaving an easily visible fingerprint behind. Substances that can leave patent fingerprints are ink, blood, dirt, flour, grease, etc. When police take inked fingerprints, they are collecting patent fingerprints.

Plastic fingerprints are 3D impressions of fingerprints left in a substance like wax, mud, paint, soap, tar, drying blood, etc. They are generally easily visible.

A good way to think of the difference is that an impression of a fingerprint left in blood is a plastic fingerprint, but a fingerprint of blood residue transferred to another surface is a patent fingerprint.

First, both types of prints are generally photographed since they are visible to the naked eye without enhancement. Different photography techniques can help to make the prints more visible in the photographs.

After photographing, patent prints are processed or lifted from the surface much as a latent (invisible) fingerprint would be. See

Plastic fingerprints are generally preserved by casting. A liquid material (silicone rubber, plaster, or a metal alloy) is poured over the fingerprint and hardened to make a cast of the impression. The cast is much more durable than the plastic fingerprint and can be stored as evidence.

References:
href="http://www.doj.state.wi.us/dles/crimelabs/p hysicalevidencehb/Ch11_LatentPrints.pdf
http://www.crimemuseum.org/library/forensics/fing erprints.html
http://fingerprintanalyst.webs.com/patentandplast icprints.htm
http://www.cengagesites.com/academic/assets/sites /4827/bertino_chapter6.pdf
http://desksgt.com/Classes/crim_inv/Week5.htm
htatps://www.ncjrs.gov/pdffiles1/nij/225328.pdf