I don’t know the “official” definitions for these terms, but here is my practical attempt at defining them.
A PUD is the term given for a development that was designed and approved in a unified manner as a single project possibly with interrelated sections (sometimes separated by phases or building types) that complies with all the municipal regulations (zoning, subdivision and possibly even PUD ordinances) and environmental requirements on such development. There can be SFR's, condos, townhouses, commercial uses, etc. in the PUD. Incorporated in the plans is the location of common utility services, open areas, recreational amenities, retention basins, etc. Sometimes these areas are publicly dedicated and sometimes they remain under the ownership of the HOA.
An HOA is typically a private non-profit organization consisting of the property owners of a given neighborhood. The HOA is usually responsible for maintenance of common areas and they charge dues to the home owners as a source of revenue. There are governing by-laws as well as regulations that create covenants and restrictions that the property owners must abide by. The HOA is usually run democratically with officers elected by the home owners as spelled out in the by-laws. Each unit owner is entitled to vote in the governing process. While under construction, the developer is often in control of the HOA, owning the bulk of the units. When the development is sold out, control goes entirely to the home owners.
The two are totally different things. When there is a PUD there is usually an HOA. However, if there are no common elements remaining (they have been deeded away) and no further need for an HOA, the HOA can be dissolved, as almost happened in my former neighborhood. Further, an HOA can exist in any neighborhood where the homeowner's desire to have one.