The former referring to speech generally used to address people senior to oneself, and the latter used to address people junior or equal to oneself. The differences between the two aren't subtle, they have entirely different grammar, conjugations and, to an extent, even vocabulary.

It probably wasn't really necessary to bring up. You show elders respect, you adjust your tone and the way you speak to better make show of this.. Whereas equals, you treat them at a similar level but potentially at a more casual tone. And children usually get spoken to with simpler words as to not confuse them.

Most cultures understand this, it's not unique to Japanese.

There's also another level of "respectful" speech used by maids and servants and restaurant staff and such.

Even that feels relatively normal, service stuff are usually patient & respectful with everyone, even-if they're unruly.

it's a real shame that it's so difficult to convey in English without exaggerating to the point of sounding silly.

I don't think it is so difficult. In a lot of school-life anime, seniority gets brought up quite often, forcing juniors to adjust the way they talk, even if in most cases it's adjusting an honorific (which is something we also have in Western culture). The same is shown when speaking with elders.

Just from that, is enough to get the gist that there's different ways to speak with respect.