A savings account has a low risk, and that corresponds to a lower return. A money market fund is also low risk, but it is still of a higher risk than a savings account. This allows for a marginally higher potential return. The return on a savings account is fixed. There is nothing you can do to increase the ROI except by maintaining a higher balance. However, in any reasonable scenario, the ROI is likely to be lower than the inflation. A savings account is a place to keep money, not grow it. With the money market fund, on the other hand, you actually have a reasonable opportunity for greater returns by undertaking a bit more risk.
Debt securities available to you via a money market fund are reasonably stable. They may include various types of government bonds and corporate bonds. They are the most reliable debt instruments. Proper weightage in your portfolio will ensure that you are almost as safe as having your funds in a savings account, but you have a higher return. Also, if you are in a developed market, the relevant securities commission monitors this market and keeps things relatively stable.
What may be surprising, is that in certain instances, the costs associated with keeping money in the money market may be equal or lower than the cost of keeping it in a savings account. Again, the fee structure has to be understood and managed. A bank will charge transaction fees, which can add up, and a fee for failing to maintain a minimum balance requirement. The fund has a management fee that is normally below half a percent. Provided you do not incur too much transaction cost for unnecessary movements of your funds in the money market, the fees can be considered negligible. The money market funds are almost as liquid as that of a savings account. There is a slight delay in converting the securities into cash, which may take a day or more. I personally consider this an advantage since it precludes impulse spending.
If the funds are substantial enough, you may incur tax. Depending on how you manage your investments, and where you put your money, and where you are domiciled, there is actually the possibility that much of what you earn in the fund would not be taxed, or taxed at a lower bracket than keeping an equivalent amount in a savings account. This might also be dependent on the vehicle used to manage the funds.