Money Market accounts are not generally insured by the FDIC except during the 2008 Financial Crisis, when President Bush temporarily extended coverage to them. Technically a Money Market is a mutual fund. It has a constant targeted net asset value of 1$ per share. But a Money Market can "Break the Buck". Meaning be worth less than what you put in. This is extremely rare. But it does happen.
In 1994 the community bankers US government money Market fund paid back .96 cents on the $1. In 2008 the Reserve Primary Fund took a sizeable loss in some Lehman Brothers paper it held and paid back less than $1 per share. It should be noted that most brokerages absorbed the loss on behalf of the clients, because if they didn't they'd loose more business in the long run. A few smaller firms passed on the loss to shareholders. Money markets are a loosing business venture in an interest rate environment of Zero short term rates. They cost more to run then they pay in interest. Firms only run them as means to gather assets so they have you as a potential customer for other services.
With a high yield savings account, the FDIC backs the loss up to 250k, which would only result in the even of an underlying bank insolvency. In general the risk is nominal for both in terms of return of principal.