The defining choice is credit, and the reward is a modest extra yield. A government money market fund lends, in effect, almost entirely to the U.S. Treasury and government agencies. A prime fund also lends to banks and companies for very short periods. That private-sector credit is why a prime fund typically pays a little more than a government fund holding the same maturities, and why it carries a little more credit and liquidity risk in return. The gap is usually small, and it can widen or reverse when short-term credit markets are stressed, which is precisely when the difference matters most.
"Prime" is a market word, not a legal category, and this trips people up. SEC Rule 2a-7 defines only two kinds of money market fund by name, a government fund and a retail fund, and describes every other fund only negatively, as one that is neither. The market calls that residual "prime" (or "municipal," when its debt is tax-exempt), but those labels are not in the rule. This matters because the widely repeated summary that "prime funds have a floating share price" is wrong for exactly the prime funds an individual is most likely to hold. A retail prime fund, defined by being limited to individual investors, keeps a stable $1.00 price. It is the institutional prime funds, open to corporate and other non-individual investors, that must price their shares to four decimal places and can be subject to a mandatory fee on heavy redemption days. The full mechanics of both belong to the money market fund page.
The 2008 crisis is why this category is regulated the way it is. In September 2008 a large prime fund, holding short-term Lehman Brothers debt that suddenly became nearly worthless, saw its share price fall below $1.00, an event called "breaking the buck." The run that followed helped freeze the short-term funding markets that companies rely on. The SEC's later reforms, including four-decimal pricing for institutional prime and municipal funds and the liquidity fees that replaced redemption gates, were aimed at this category, because a prime fund's private-sector holdings are what can lose value in a credit panic. A government fund faces far less of that particular risk, which is why the two categories are treated differently.