What is the difference between grasshoppers and crickets?

It started with an innocent Google search to try to identify the common flying crickets/grasshoppers that we see around here (I'm pretty sure they are Carolina grasshoppers). But then I went down a rabbit hole, as happens so often, over the difference between crickets and grasshoppers.
I was always under the misbegotten impression that grasshoppers are green and crickets are brown (which is why I always thought that Carolina grasshoppers were actually crickets). But of course, like the whole butterfly/moth dichotomy, it's nothing like that simple. It never is. Anyway, here goes.
Both crickets and grasshoppers are part of the insect order Orthoptera, which covers grasshoppers, locusts (which are actually just grasshoppers that have a migratory phase), crickets and katydids. Grasshoppers and locusts are in the sub-order Caelifera, while crickets and katydids are in the sub-order Ensifera. So, grasshoppers and crickets are closely related, but as cousins rather than siblings, so to speak.
They both have those long back legs that allow them to jump huge distances, but there are various differences between the two, as a picture of a generic grasshopper and a generic crickets shows:

A grasshopper is typically bigger and has a more rectangular body, while a cricket has a more streamlined (teardrop-shaped?) body, and many look almost beetle-like. Grasshoppers have relatively short antennae, while crickets have long antennae, often as long their entire body.
Other differences? Grasshoppers are mainly herbivorous and can be quite a pest for farmers, especially when  they swarm in their billions like African locusts; crickets are mainly predatory or at least omnivorous. Grasshoppers are mainly diurnal (active during the day); crickets are main nocturnal. Grasshoppers "stridulate" by rubbing their hind leg against their forewing, producing a kind of buzzing sound; crickets rub their forewings together, to produce a more chirping or trilling song. And, if you happen to get close enough, grasshoppers have their auditory organs on their abdomens, while crickets have theirs on their forelegs, and grasshoppers have short ovipositors compared to the long, extended ovipositors of crickets.
And yes, most grasshoppers are in fact green (although some are not), and most crickets are brown or a paler green (although some are bright green). Oh, and most types of grasshoppers do actually fly, not just hop (they use those huge, strong back legs to push off before they start to use their wings); but most species of crickets also fly, although some species fly much better than others, and some are short-winged or actually wingless and therefore hop rather than fly. Confused yet?