Post consumer regrind (PCR) is Plastic (mainly HDPE, LDPE and PET) that has been cleaned and reground into pellet or flake form to make a new plastic object (packaging is a big one). So in short post consumer regrind or PCR is recycled plastic.
PLA is short for Poly lactic acid and is one of many bioplastics. This one in specific is made from corn starch. Not sure but I think Natureworks is the main supplier.
The difference? PCR is more stable and has many more uses because the material is just recycled plastic. PLA, because it is made from natural substances, breaks down, melts and losses its rigidity a lot faster and is typically better for one time use. PLA is best used for products that will not have to endure a long shelf life or hold volatile products.
There are many advancements being made to bioplastics/biopolymers that will improve its rigidity, strength and shelf-life. It's hard to keep up! PLA is only one of many biopolymers. They can be made from starches such as potato, pea and corn as well as sugar. But I'm sure there are many more in development!
And now for my shameless plug: Packaging Logistics offers a wide range of beautiful packaging in 100% PCR PET! The material is FDA approved and minimums start at 50,000 pieces. Consider converting your current PET packaging to PCR PET! You'll be doing the environment a favor!