Consider the following two snippets, with braces:
switch (var) {
  case FOO: {
    x = x + 1;
    break;
  }
  case BAR: {
    y = y + 1;
    break;
  }
}
<         
        You say the braces can be omitted because no variables names are being reused. By reusing variable names, I assume you mean declaring a new variable of the same type.
The braces are actually most useful to ensure you don't end up reusing the same variable in different cases by mistake. They don't declare any variables today, but someone will add them tomorrow and without the braces the code is error-prone.
With braces.
There are so many things that can go wrong with switch statements I try to avoid them where I can, i.e.
Using braces is one way to prevent both intentional and accidental sharing of variables between case statements