I came across this really informative post on this topic about a year ago. Credit to Shi Zhong for writing it!
MMM can be viewed as one type of attribution modeling that started from and focused more on traditional (offline) media channels, where no granular user-level data is available (this could change in the future). It can give you at a high level what portion of your total sales was driven by each media channel. It normally require a very long history of data to gather enough information for modeling.
The "new" Attribution Modeling seems to have started more from digital media and focused on leveraging user-level (cookie-level) data to figure out conversion credit deserved by different media channels/campaigns/placements/keywords (groups)/etc. At user-level, one can construct the exact sequence of touch points leading up to a conversion. By analyzing all such sequences, one can derive the fractional conversion credit for each touch point on the conversion sequence. It has gotten popular recently as more and more (digital) advertisers and agencies realize the previous de-facto last-click attribution method (used by all ad servers) cannot accurately capture the contribution of different media channels/campaigns/placements, especially for upper-funnel channels such as regular display. Display channel can help drive users down the conversion funnel and convert through Search channel but Search channel would normally take most of the credit according to last-click attribution method.
MMM is needed when one does not have user-level data. For example, in order to consider digital channels and offline TV media together, we have to join the data at some aggregate level and use an MMM-type approach. Traditionally another main use of MMM (sometimes also called MMO, marketing mix optimization) is for forecasting future sales.
User-level Attribution Modeling is preferred when you have user-level data as models based on user-level sequences can be more accurate and more causal.