I'm no expert and excuse my roughness but my understanding is as such:

The fed seems to believe that since it's all loans (qe) and will theoretically be paid back, even at "unlimited" levels for the immediate future, long term inflation expectations will not be shifted upwards as markets will understand it is temporary and exists only in response to the crisis.

MMT is about using immense amounts of fiscal policy from government funds and directing central bank decisions? In theory I think this is the big difference. MMT believes they will be able to use fiscal Policy similar to how the fed does, and take out money through taxation to again not incur ramped up long term inflation expectations. Fiscal Policy though is inherently different I think (?) In how it interacts with markets than monetary policy. Often monetary policy doesn't work for everything, but fiscal Policy would be powerful enough (utility optimization not considered). This power might have a drawback of driving long term inflation expectations upwards. Either expectations for long term increases in government spending for programs (and they all tend to get more expensive with larger populations right?) Drive belief of increased money supply driven by "guaranteed " government spending.

Even if Political power changes and spending is reduced, the "promise" of removing the money from the economy through taxes is not "guaranteed" unlike with loans from the federal reserve. The removal of money from the money supply through fiscal Policy is inherently tied by political winds. Those haven't always been the most predictable and certainly not reliable, unlike the fed getting it's loans repaid. This suggests to me again that MMT will suffer from possibly "real " inflation due to fiscal Policy being stronger but more blunt than monetary policy and it being much more likely the "temporary" increase in money supply becomes permanent.

Rough references

https://en.m.wikipedia.org/wiki/Trillion-dollar_coin

Inflation risks parts has fed talking about why it sees it's qe money operations as effectively inflation neutral

https://en.m.wikipedia.org/wiki/Modern_Monetary_Theory describing the increased role of elected govt in monetary decisions for fiscal Policy means and relying on taxation as another levy of money out