The number one difference is that bank savings accounts, or money market accounts (MMAs allow limited checking--six non-ATM withdrawals per month, max, else possible fees) have FDIC insurance up to $250,000. However don't put that much in--allow some room for interest, so you never go over the $250,000. Money Market Mutual Funds do not enjoy FDIC insurance. There may be some SPIC insurance--generally against brokerage failure though, but its coverage is questionable--search out those details, and if they apply to anything besides actual cash held at the brokerage. If the money market mutual fund is strictly invested in US Treasury securities (like T-Bills, or other short-term US Treasury instruments), it enjoys the full faith and credit of the US government, FWIW--but many MMMFs invest in corporate instruments. If the fund has any pricing issues, there might be a delay in getting paid off. (Extremely unlikely.)
Number two, and more importantly, bank savings accounts (or MMAs) pay way more! You can get a bit over 1% APY now--many paying 0.90% APY, or higher. No money market mutual funds are close to that, generally yielding a small fraction of that, almost zero for US Treasury MMM funds. Sure 1.05% ain't too exciting, but you may as well get the most you can if holding "cash," and fully insured to boot.