Short version: PTAs are formally affiliated with the national PTA (and its state affiliates). It's a national membership organization, which requires dues and provides services and benefits.
"PTO" (parent teacher organization) is the general acronym for all the other parent groups. These often have acronyms like HSA (home and school association) or PTC (parent teacher council) or PTO, but they all just mean indepependent parent groups.
Roughly 75% of the parent groups in the US are *not* affiliated with the PTA and are considered PTOs. To become a PTA, you'd start by talking to the PTA at www.pta.org