Difference between Riba-based Banks and Islamic Banks
Question

If Islamic banks do not use the interest-based system, then what benefit do they gain and how can that help them? Is the fee they charge in return for their service regarded as being similar to Riba? What are the transactions that Islam regards as being Riba?
Praise be to Allah.How Riba-based banks work
The interest-based system on which commercial banks are based is a prohibited system that is based on lending and borrowing with Riba. The bank lends money to the customer with interest, and the customer who deposits money in the bank lends this money to the bank in return for interest. Loans with interest constitute Riba , regarding which there is scholarly consensus that it is prohibited. 
How Islamic banks work
Islamic banks are based on permissible transactions such as selling, buying, profit sharing, partnerships and other Islamically-acceptable forms of investment, in addition to the fees charged for wire transfer, benefiting from variations in currency exchange, and currency exchange.
Difference between Riba-based Banks and Islamic Banks
There follows a straightforward example to explain the difference between Riba-based transactions and Islamically-acceptable transactions, and how the bank benefits from doing either of the two transactions: 
If the customer wants to put his money to good use and make it grow, then he deposits it in a savings account in the Riba-based bank , and the bank allocates to him a known percentage of interest, whilst guaranteeing his capital. This in fact is a Riba-based loan, a loan from the customer to the bank. The benefit to the bank is how it uses the deposited money , which it lends to another customer in return for interest to be paid by the customer. Thus the bank borrows and lends, and benefits from the difference.
As for Islamic banks, one of the ways in which they invest is to take money from the customer and invest it in a permissible business, or to set up a housing project and the like, on the basis that it will give the customer a percentage of the profits, and the bank – as the entity that does the actual work – will also have a percentage of the profits. Thus the bank benefits from the percentage that it takes of the profits generated by the project, and its share of the profits may be much greater than what Riba-based banks collect from Riba, which is prohibited. But in the case of profit-sharing there is an element of risk, and the bank has to try hard to select a project that is profitable and monitor it until it bears fruit.
The difference between the Riba-based bank and the Islamic bank in these examples is the difference between prohibited Riba-based loans and Islamically-acceptable profit-sharing in which the customer may lose his money, because there is no guaranteed protection of his capital, but if he makes a profit, this gain is a lawful wealth.
The point is that the Islamic bank has many acceptable ways of making a profit, and hence these banks have begun to grow and flourish. In fact some non-Muslim countries are trying to apply the Islamic banking system, because it makes a profit and avoids much of the mischief that stems from the Riba-based system and is the cause of ruin and loss.
Types of Riba-based transactions
Riba-based transactions are of many types, such as lending and borrowing with interest; currency exchange (selling one currency for another) and deferring hand-to-hand exchange; exchanging gold for gold in different quantities, or with the exchange to happen later; transactions which basically boil down to being Riba-based loans, such as discounted bills, savings accounts, investment certificates with returns or prizes, charging penalties for late payments in the case of sales by installments or withdrawing cash using a credit card. 
And Allah knows best.