I actually just recently assigned an article in my North American archaeology course on part of this subject. If you haven't read Sarah Sherwood and TR Kidder's 2011 paper, The DaVincis of Dirt I recommend it. In the meantime, some highlights...

First, moundbuilding in eastern North America is / was an ancient tradition. Purposeful, intentional mound construction appears as early as 5600 BP in northern Louisiana at Watson Brake, and the expansion of moundbuilding as a cultural tradition seems to spread from that region. By the time of Poverty Point and the culture(s) that seem to have constructed that site, moundbuilding would already have been an ancient tradition. The development of these types of monumental architecture in the Late Archaic of eastern North America is every bit as impressive and "out of place" as Gobekli Tepe was in its region: large, monumental constructions erected by hunting and gathering peoples over, in some cases, very short periods of time. TR Kidder has estimated 3 months from start to finish for Mound A at Poverty Point, one of the largest earthen mounds-- and the largest Archaic mound-- in North America. The estimate derives from stratigraphic and geoarchaeological evidence from the mound itself. Internal composition shows no major erosional episodes. In Louisiana, it's hard to go more than a few weeks without rains. One of the only times that's possible is during the summer, and so Kidder has estimated that based on that evidence (and other indications-- see "Poverty Point and the Archaeology of Singularity" for a quick recap), the mound was built-- by hunter gatherers, no less-- over a very short timespan.

Later moundbuilding peoples left conical burial and effigy mounds-- the Adena and Hopewell peoples of the Woodland period, for example. And they left some amazing examples. Great Serpent Mound in Ohio, for example.

However, the largest and most impressive mounds in terms of scale and engineering seem to be the Mississippian-period mounds, ca. AD 900 to roughly 1500. Relatively little is known about the construction of burial mounds for the Mississippians, because the types of sophisticated analyses that Kidder and Sherwood did for Mississippian and Archaic mounds are not possible with burial mounds. Federal legislation has made it very difficult to get permission to excavate burial mounds (note: I do not have a problem with that, please don't cloud the comments with bitching about NAGPRA).

Mississippian platform mounds need to be seen as two things:

1. Feats of major engineering every bit as impressive as the Great Pyramids, requiring sophisticated understanding of soil types and properties. (After all, many of these edifices are still around, and largely intact.)

2. The end products of multiple episodes of community building, and the crystallization of social relationships in those communities over multiple generations.

For the first... extensive excavations at sites like Cahokia and Shiloh Mound A have shown that Mississippian mound construction was elaborate, careful, planned, and most importantly, engineered. Clays and other sediments were chosen for their physical properties and in some cases mixed to achieve careful internal structures that promoted stability while also maintaining drainage to prevent water "perching" on clay deposited in the mounds. Rather than "pile of dirt," think "earthen dam" or "artificial flood levee."

It's likely, based on the internal structure and clear intent / planning involved, that Mississippian mounds were built (and the labor directed) by the equivalent of Mississippian engineers and architects.

Furthermore, the excavations-- especially at Shiloh-- show us that contrary to depictions of grass-covered mounds-- Mississippian mounds were not grass covered at all during their use. Rather, they were well-maintained, covered in colored clays-- reds and yellows-- that required constant and careful maintenance. Organic soils (as you would need to get growing grass) are dark, and in a profile it's easy to pick out soils that were buried by later deposition. They generally create a dark linear band beneath later deposition. There is no such evidence of that in (for example) profiles from Shiloh's Mound A (see below for the link to the page with more images).

So Mississippian towns and cities would have not looked quite so "pastoral" in character when they were occupied. Instead, imagine something like a combination of a European walled town-- most Mississippian towns were surrounded by a palisade consisting of a log core plastered over, the result being a fairly substantial and effective defensive wall-- and a Mesoamerican city, complete with one or more brightly colored pyramidal mounds surrounding a well-maintained open plaza.

Second... mound building was a community effort. It was the journey that was important, not just the mound to put stuff on top of. Think "barn raising" among the Amish. Mississippian mounds were constructed in stages, with individual stages being used for periods ranging from 10 to 40 years, give or take. It's been hypothesized that when the important person occupying the structure atop a mound died, or when power changed hands, the structure was burned and a new mound stage was erected over the mound, raising it higher and rejuvenating both the mound and (in the community effort to raise the new stage) the community.

So, to the specific questions:

Is there a formal distinction between these types and do they have distinct characteristics/construction methods, or do the functions often overlap?

In some cases, yes. Although Woodland period mounds appear to have been ritual and mortuary in purpose, Mississippian towns contain both burial mounds and platform mounds. Some Mississippian burial mounds appear to be conical in form, and do not seem to have the space at their summit for structures. Others are very clearly platform mounds in various configurations, intended to have one or more structures or public open areas atop them.

There is overlap. Some flat-topped pyramidal mounds, like Mound C at Etowah in Georgia, were also burial mounds for seemingly important members of the society. In these cases, the structures atop them are thought to have been charnel houses (basically Mississippian funeral homes) where the dead were carefully treated and prepared for burial. Although (for reasons mentioned above) we don't know much about the internal structure of these mounds, we can surmise that since they were built in part to support summit structures and activities, they were likely constructed in much the same way as non-burial pyramidal mounds.

Were burial mounds more common in some regions or times?

Mounds strictly for burial only were more common during the Woodland period in the eastern US. By the Mississippian period, we see burial areas in and around some mounds that were intended for certain members of society, but many other burials in other areas of towns, including in house floors. By contrast, we see little evidence of platform mounds during the Woodland period, and mounds are largely burial or effigy in nature.

Were mounds repurposed as burials?

Yes. Some Mississippian sites (e.g., Moundville in Alabama) show some evidence of intrusive burials in abandoned pyramidal mounds that were likely originally intended as substructure mounds, and don't appear to have originally been intended as burial mounds. The reasons for this are thought to be similar to the reasons that other peoples around the world have often re-used ancient but clearly significant sites for burial-- interment of the dead in a sacred place, incorporation of the dead into the legends about the people who occupied those ancient spaces, etc.

The NPS's photo catalog from the early 2000s Shiloh Mound A excavations.