Why are United States coins different sizes?

For two major reasons.

First, when the U.S. started issuing coins in the 1790s, by law a coin had to contain metal whose value was the same as its denomination minus a small amount to cover minting costs. That meant coins had to contain different amounts of metal, or different metals altogether. Cents were large, about the size of a half-dollar, because they had to contain about 0.98¢ worth of copper. Similarly a dime had to contain about 9.9¢ of silver, etc. etc. That meant that coins made from the same metal were sized in proportion: a quarter was 2.5X the size of a dime, a half dollar was 5X the size, etc. It's also the reason that a dime is smaller than a nickel or a dime, because silver was more valuable than copper or nickel.

As the prices of metals changed over the years the country gradually got away from mandating that coins be worth as much in metal as their denominations. Cents were downsized to their current diameter, the nickel was introduced, and so on. Eventually by the 1960s silver was completely eliminated. However, by that time there were so many vending machines and coin-counters that it would have been extremely disruptive to change sizes again, so coins were kept the same size as before. The new dollar coins were the only exception because there were no machines that took the old coins; the Mint was able to downsize that denomination without forcing major retrofits.

That leads to the second reason. If all coins were the same size and color, it would be very difficult for people to tell them apart! For example, in 1943 the Mint was forced to make cents out of steel because of wartime copper shortages. The coins were silver-colored and a lot of people accidentally spent them as dimes! You can also read about the lengths the European mints went to when designing the new euro coins - each one has a special size AND edge to help prevent confusion.