Richer areas often newer areas. Not always, of course, there's plenty of "old money" areas but you also see plenty of cases of new development particularly for middle and upper middle class. They want nice new homes, those homes are built in new developments.
Now why's that matter? Well when you are building a new development, you usually use the most current technology which often means FTTH, or at the very least higher quality category cable and fiber out to the box. That lets them offer higher speed. The big cost is running the lines, not the material used so you do it with better materials. You have to spend the money to lay the lines, or you can't offer service.
However in old development, well that has old shit. It can be replaced, of course, but that is a lot of money. It can cost more than a new run because tearing shit up in a developed area can be pretty costly. So they are reluctant to do it.
This of course goes double if you are talking areas that are poorer. The improved infrastructure would allow them to offer faster speeds, but the reason they want to do that is because they can get more money. People who live in poorer areas are not as likely to want to spend more money and will just elect to keep slower speeds. A good number of them might not even be on the fastest speed available to them already because they wish for something cheap.
Thus it makes sense why it happens like that. The reason cable companies offer faster speeds is it is generally much easier for them particularly with DOCSIS 3. All they really have to do is put more channels on their CMTS. It isn't free, but doesn't cost a ton and doesn't require redoing lots of buried cable. The coax out there is already good to a gigahertz, maybe more.
You even see it in middle class neighborhoods. I live in a decent condo complex, and right next to me is some pretty upscale housing. However, both here and in the houses, 6ish mbit DSL is all you can get. Reason is it is old construction, 1970s. So the telephones are all copper, straight to the CO, and not very high grade cable. The cable company will sell you 300mbit though, no problem. That said the same cable company offers fiber in new developments, many of which cost less than the houses near me.
It is just what we are going to see with for profit companies. If we want an "equal speeds for all, don't worry about the costs" setup then it is going to have to be publicly funded and run.