Great question. There are several reasons; I'm going to list the few that I can think of off the top of my head right now.
First, even if institutional bank holdings in such a term account are covered by deposit insurance (this, as well as the amount covered, varies geographically), the amount covered is generally trivial when seen in the context of bank holdings. An individual might have on the order of $1,000 - $10,000 in such an account; for a bank, that's basically chump change, and you are looking more at numbers in the millions of dollars range. Sometimes a lot more than that. For a large bank, even hundreds of millions of dollars might be a relatively small portion of their holdings. The 2011 Goldman Sachs annual report (I just pulled a big bank out of thin air, here; no affiliation with them that I know of) states that as of December 2011, their excess liquidity was 171,581 million US dollars (over 170 billion dollars), with a bottom line total assets of $923,225 million (a shade under a trillion dollars) book value. Good luck finding a bank that will pay you 4% interest on even a fraction of such an amount. GS' income before tax in 2011 was a shade under 6.2 billion dollars; 4% on 170 billion dollars is 6.8 billion dollars. That is, the interest payments at such a rate on their excess liquidity alone would have cost more than they themselves made in the entire year, which is completely unsustainable.
Government bonds are as guaranteed as deposit-insurance-covered bank accounts (it'll be the government that steps in and pays the guaranteed amount, quite possibly issuing bonds to cover the cost), but (assuming the country does not default on its debt, which happens from time to time) you will get back the entire amount plus interest. For a deposit-insured bank account of any kind, you are only guaranteed (to the extent that one can guarantee anything) the maximum amount in the country's bank deposit insurance; I believe in most countries, this is at best on the order of $100,000. If the bank where the money is kept goes bankrupt, for holdings on the order of what banks deal with, you would be extremely lucky to recover even a few percent of the principal. Government bonds are also generally accepted as collateral for the bank's own loans, which can make a difference when you need to raise more money in short order because a large customer decided to withdraw a big pile of cash from their account, maybe to buy stocks or bonds themselves.
Government bonds are generally liquid. That is, they aren't just issued by the government, held to maturity while paying interest, and then returned (electronically, these days) in return for their face value in cash. Government bonds are bought and sold on the "secondary market" as well, where they are traded in very much the same way as public company stocks.
If banks started simply depositing money with each other, all else aside, then what would happen? Keep in mind that the interest rate is basically the price of money. Supply-and-demand would dictate that if you get a huge inflow of capital, you can lower the interest rate paid on that capital. Banks don't pay high interest (and certainly wouldn't do so to each other) because of their intristic good will; they pay high interest because they cannot secure capital funding at lower rates. This is a large reason why the large banks will generally pay much lower interest rates than smaller niche banks; the larger banks are seen as more reliable in the bond market, so are able to get funding more cheaply by issuing bonds.
Individuals will often buy bonds for the perceived safety. Depending on how much money you are dealing with (sold a large house recently?) it is quite possible even for individuals to hit the ceiling on deposit insurance, and for any of a number of reasons they might not feel comfortable putting the money in the stock market. Buying government bonds then becomes a relatively attractive option -- you get a slightly lower return than you might be able to get in a high-interest savings account, but you are virtually guaranteed return of the entire principal if the bond is held to maturity. On the other hand, it might not be the case that you will get the entire principal back if the bank paying the high interest gets into financial trouble or even bankruptcy. Some people have personal or systemic objections toward banks, limiting their willingness to deposit large amounts of money with them. And of course in some cases, such as for example retirement savings, it might not even be possible to simply stash the money in a savings account, in which case bonds of some kind is your only option if you want a purely interest-bearing investment.