I've found in my climate that red and black pines respond well to similar care. I give both full sun and it doesn't get cold enough for either to require protection in winter.
I tend to keep red pines slightly drier than black pines. I've seen red pines yellow with too much water but green up when kept a bit drier. They're also the first pines to wilt when they get too dry.
This goes for soil too - I'll usually use a bit more akadama for black pines than for red.
Red pines often have more slender needles with a bit of a twist. The shoots can be thinner and the needles a more pale green color. The buds are red while black pine buds are white. Red pine bark can be flaky or plated and sometimes has a ruddy color.
In terms of growth characteristics, I've found black pine growth is steady and predictable, whereas red pine growth can respond to training with greater variability. Cutting or plucking too much can produce long, lanky shoots on red pine, and over stressing red pines can result in no summer shoots at all. The same is true of black pines, but these effects are more often seen on red pines.
I think of the biggest differences between the two as being stylistic. Red pines are frequently trained with slender trunks in upright styles that have a bunjin feeling. Sharp angles are more common in red pine bonsai. Red pine are sometimes trained to have large trunks with fast taper, but this approach is far more common to black pines to red.
My understanding is that these stylistic differences reflect the ways the trees grow in nature, with collected specimens exemplifying the styles with which we're familiar. Red pines grow up into the foothills and higher while black pines are more commonly shore trees. Both varieties usually take tall, slender forms in nature.
This is a start - I'll update this as other distinctions come to mind.