In general, the golden rule of micro-optimization applies here:
"Measure, don't guess".
As with all optimizations of this type, the difference between the two calls almost certainly won't matter unless you are checking millions of strings that are each tens of thousands of characters long.
Run a profiler over your code, and only optimize this call when you can measure that it's slowing you down. Till then, go with the more readable options (startsWith, in this case). Once you know that this block is slowing you down, try both and use whichever is faster. Rinse. Repeat ;-)
Academically, my guess is that startsWith will likely be implemented using indexOf. Check the source code and see if you're interested. (Turns out that startsWith does not call indexOf)