From the British and International Golf Greenkeepers Association, Master Greenkeeper Certificate, module examination syllabus:
"Heathland
"These courses have been formed in an artificial habitat, created many centuries ago, by clearing natural forest and introducing grazing. It was on these sites to the south and west of London that golf really became popular over 100 years ago, since the land offered was considered the nearest to traditional links. The soils tend to be poor and dry, often sandy in nature with a low pH. Rainfall also tends to be on the low side, and as a result, growth in general is fairly minimal. Fine-leaved indigenous bents and fescues predominate along with bell and ling heather and perhaps gorse and broom. Scrub birch and pine tend to encroach and a carefully balanced policy of conservation, not preservation is vital for the survival of these diminishing tracts of land."
and later in the text:
"Moorland
"They can often be described as upland heaths where low pH peaty soils have dried out or where acid mineral soils exist. These areas, mostly in northern England and Scotland were once covered by great tracts of forest. Many courses in this environment are treeless, windswept, and exposed to the elements, often with high levels of rainfall. The growing season is limited and snow often covers much of this land for several weeks in the year. Plant cover is dominated by the acid-loving heathers and bentgrasses along with wavy hair grass. Sedges may also grow when soil conditions are wetter."
The syllabus goes on to explain general course management strategies for each, and similarly covers parkland and links courses, as well.
So, according to the British greenkeepers, who should know, the two courses are similar with acid soil, bentgrass, and heather, but they differ in that the moorlands are colder, wetter, and have more organic matter or minerals in the soil, and the heathlands are more likely to be invaded by trees such as pine and birch.