I am a bit confused right now about what a hedge fund is. If Investment Banks can invest the money entrusted to them by clients, what makes someone want to invest with a hedge fund?
Isn't a particularly clean answer to this question, but roughly:
Investment banks generally are sales entities, raising and selling corporate debt to clients who want to buy/sell bonds etc, and working on rights issues/ipos and selling stock directly to people who want to own parts of companies. These are (mostly) singular transactions that are done and moved on from, where the bank doesn't hold positions for long.
They often do also offer wealth management, but again mostly as a broker, selling funds/stocks etc to clients but again not taking much/any risk and just charging broker fees on these transactions.
'Hedge fund' is a very loose term and can cover a lot of different things, but they are generally much more long term, market directional bets on companies and debt where the owners usually have considerable holdings of their own funds in these directional bets, and charge performance related fees depending on how the fund performs.
A vague analogy would be that investment banks are generally like supermarkets, taking in goods and quickly selling them to customers, where as hedge funds are usually much more like entities that make longer term investments in new and existing products that shops can then sell, which carries a quite different risk profile and investing horizon.