I think of wealth mgmt as something closer to financial planning, not something like research. So you'd be working one-on-one with rich folks, helping them manage their portfolios, account for taxes, handle estate planning, that sort of stuff. So in some cases it would be the retail side of IM, but there are also independent shops.
For investment mgmt, I'm talking about working as a research analyst for a fund, perhaps covering 15-20 companies in a certain industry or being more of a generalist. So you're doing things like looking through SEC filings, building financial models, listening to earnings calls, etc, all in an effort to value the stock.
The thing is, investment mgmt is a very scalable business, so these funds can take on tons more assets without adding new people. They make the same amount of decisions, they just buy bigger positions. For example, I've been talking to a well-respected mutual fund with over $200bn in assets. They only take 2 interns per year. Makes you want to resort to i-banking