Gotham is focused on displaying entity based data. Entity based data was often things like people, reports of events, invoices, etc. Gotham let you do things like click on a person, and see all criminal records associated with them. Or their relatives and known associates. This was called "search around" and was apparently a very big deal. One sample use case for Palantir Gotham was to search for people that made more than one purchase of the same Schedule 2 drug, and plot those purchases on a map. A common method of finding pill mills was to plot these drug purchases on a map and see when there's a bunch of purchases along a highway route. Big indicator that someone is buying in bulk, but splitting up their purchases to fall under thresholds which traditional alerting mechanisms relied upon. Palantir Gotham is the product that people usually associate with Palantir - it's the one used by a lot of three letter agencies.
I don't know too much about Metropolis. It was more about quantitative analysis. I think it was popular for insurance companies. IIRC it could do things like plot the frequency of adverse weather events on a map, and then insurance adjusters to fine tune premiums. I think there was also a Palantir Metropolis product aimed at small businesses.
Now, the above covers the frontend. A huge part of Palantir's operations is data ingestion. Basically, this consisted of moving data out of government or enterprise databases and moving them into Palantir's format. This usually involved forward deployed engineers that would work with the government or business and work out a way to ingest the data. An un-glamorous job but one of the really important ones.
Another huge component of the business was customer specific customization. When I worked there something like 40-50% of software engineers worked on business development. They rest worked either on infrastructure of UI platform (basically building the components of the UI that aren't customer specific). Business development at Palantir when I worked meant "software development for a given business (or agency)" not sales. If you worked in business development you were a developer for a given business or agency. This was also a lot of grunt work. One customer wanted to bucket items based on the phase of the moon. I distinctly remember this ask - maybe the FBI was investigating a werewolf. Implementing features and customizations like that was the role of business development.
Is it a powerful tool or is it snakeoil? Well, given that the workflows it's replacing are often whiteboards and excel spreadsheets it may actually be a very significant gain. Ultimately what Palantir has is a flexible way of displaying and finding relationships between data. Flexible enough to ingest very different types of data. Perhaps more importantly they have a technical workforce capable of rapidly extending the product to meet new customer needs. And if customers are happy with the product I guess it's worth it.
I just hope they've moved off of god-damn java swing for the UI.