First things first, it is important that you realize that you can get a Health Savings Account from literally any bank that offers them, not just entities that have manipulated their initials to be very close to HSA.
Okay, now to answer your question. It looks like Health Savings Administrators offer you a service (and charge you for it) that enables you to invest money in your HSA account into 22 Vanguard funds they have selected. So the interest you'll receive is whatever those funds end up paying.
HSA Bank looks like it's roughly the same thing but with many more options, allowing you to invest your money either with TD Ameritrade or DEVENIR Mutual Funds (whatever that is). Again they charge fees, and again the interest will depend on what you choose and how well it does, same as any other stock / mutual fund investment. HSA Bank is set up so that you'll have two accounts, a cash savings account and an investment account. I suspect that the $5k minimum to avoid fees is the sum of your money in both, but you'd only be getting interest / capital gains on money in the investment account.
This setup is not unique to these two organizations. My HSA at work is through Chase Bank, and then offer a similar investment account with a selection of mutual funds available. I pay no fees at all now that I meet the minimum balance, although that could be part of a deal that my company has worked out with them.