What benefits do depositors (savers) and borrowers obtain from banks that make it worth the extra cost?

First, banks match up savers who want to lend funds for short time period with borrowers who want to borrow funds for long time period. Another way to say this is that financial intermediaries are in a position to borrow short and lend long. Banks, for example, effectively borrow funds from savers when they accept savers deposits. These deposits are often redeemable on demand, or at least on relatively short notice. Funds in checking accounts, for example, are redeemable on demand.
In contrast, banks often loan funds for fixed terms, and these terms may vary from months (car loans) to years (home mortgage). The bank performs the valuable function of converting funds that savers are willing to lend for only short periods of time into funds that the bank itself is willing to lend to borrowers for longer periods. Of course, a bank could get in severe trouble if a depositor demanded redemption of her or his demand deposits while the deposited funds were loaned out in a 15-year loan. If unable to come up with the funds to redeem the demand deposits, the bank would be illiquid.
Banks are able to avoid illiquidity while borrowing short and lending long by using several business practice. First, a bank seeks a widely diversified set of depositors so that no one depositor is likely to cause a liquidity problem by withdrawing funds on short notice. Also, with a large number of depositors, the odds are that on most days the dollars added to some accounts and withdrawn from others will roughly cancel each other out, leaving total deposits fairly stable. Second, a bank keeps a small amounts of deposits as reserve against sudden withdrawals and, more important, establishes several lines of credits with other banks and with government regulators (e.g., the Federal Reserve System). These lines of credit can be used to meet unexpectedly large withdrawals of deposits. Third, a bank makes long-term loans to only a small proportion of its loan clients. For example, banks make Avery small proportion of housing loans relative to their total loan portfolios.
Finally, a bank "lends" a portion of its funds by purchasing government bonds, which are relatively easy to convert into currency. In this way, a portion of the bank's loan portfolio is kept in highly liquid assets, which can be sold to obtain funds to meet unexpected withdrawals.
The second valuable function banks perform as intermediaries in financial market is pooling many small deposits to make relatively large loans to borrowers. If you have only $ 1000, you cannot make a $100,000 loan to someone seeking a mortgage. But with a bank serving as an intermediary, you and other individuals can deposit your small sums in a bank to earn interest. The bank, in turn, can pool the numerous small deposits into a sizable quantity of funds to make the $100,000 mortgage.
The third function bans serve is helping depositors reduce risk. Small depositors, and even relatively large depositors, often find it difficult to minimize the risk they face in lending funds directly to borrowers, if Natalie lends her entire life's savings to a single borrower who defaults, she loses everything. While this may be an unlikely occurrence, it is not impossible. An alternative for Natalie is to lend out her life savings in smaller amounts to a large number of borrowers so that if one borrower defaults, all is not lost. This alternative is called diversifying: the saver lends his or her funds to a diverse group of borrowers so that all is not lost if one or even several borrowers default. It is difficult to engage in diversification without a financial intermediary; borrowers typically want large sums of money, and small depositors usually cannot supply large sums to a diverse group of borrowers. Thus, banks provide a way for savers, especially small savers, to diversify. Each depositor bears only a small risk, since default by one or even several borrowers will have relatively little impact on the bank or its depositor.