All the pitfalls of concurrency are there - in particular when executing non-idempotent functions multiple times before previous executions finish, then you need mutexes!
If you need to synchronize stuff in the program you can use normal plain variables, since it's guaranteed that your task will be never interrupted till you give control back to the scheduler by performing an await operation.
In a way, async code can be used to implement mutex (or something similar) themself: it's a technique that I use often in JavaScript, to implement stuff that works like a mutex or a semaphores with just promises to syncronize stuff (e.g. you want to be sure that a function that itself does async operations inside is not interrupted, it's possible to do so with promises and normal JS variables).
await foo()
await bar()
Please don't implement this yourself
This isn't even remotely true; plenty of languages have both async and concurrency, probably more than ones that don't. C# was the language that originated async/await, not JavaScript, and it certainly has concurrency, as do Swift, Python. Rust, and many more. You're conflating two independent proprieties of JavaScript as language and incorrectly inferring a link between them that doesn't actually exist.