One potential advantage of a Map is that its keys can be of any type - unlike plain objects, in which a key may only be a string or a symbol. Something else to consider is that plain objects with dynamic keys can rarely be problematic due to property name collisions (__proto__, valueOf, etc).
That said, in React, a Map probably isn't that great of an idea:
In React, state should never be mutated, but if you use a Map, you'll probably be very tempted to use methods like Map.set - which mutates it. You'll have to remember to clone the map first every time.
You won't be able to do
setState({ ...stateObj, [prop]: newVal })
but instead
setState(new Map(stateMap).set(prop, newVal))
Iterating through Maps is somewhat cumbersome because the only way to do so is through their iterators (.entries, .keys, .values) - but iterators aren't arrays and can't be .mapped to JSX immediately. Every time you want to do something with the Map, you'd have to turn it into an array instead: [...myMap.entries()].map((key, value) => which is a bit ugly.
If dynamic keys are something to worry about, you can use an array instead: const mapper = [{ key: 'foo', value: 'bar' } ... without having to use a Map.
It's not that a Map can't be implemented properly, or that a plain object can't either - you just have to be careful.