The differences between the 2 accounts are well defined by Federal regulations and vary even between the type of charter through which the establishment has formed. A Money Market account at a Credit Union varies greatly from one at a Brokerage house (i.e., Merrill Lynch,UBS, etc.). Money Markets were introduced to Banking in the 80's and 90's as Financial Institutions were deregulated for all practical purposes. Theoretically the MMA's (money market accounts) were supposed to earn or pay a higher interest rate or "rate of return" than a "passbook" savings account and allow the differing financial entities to compete on an equal footing. The restrictions placed on the account are there so the financial institutions can maintain a better grasp of how much of their clients deposits will be on the bank's books so the bank can match the timing of when those funds might be withdrawn with the maturities of the investments the bank makes to earn profit and pay the specified rate of return to the customer.
They are more or less the same but a money market deposit account is considered a savings account for only some purposes upon which checks can typically be written, subject to certain restrictions. The typical restrictions are that a fairly high minimum balance must be maintained in order to avoid fees.
Here's a great article that compares the advantages and differences of a money market account and a savings account.