Are there any practical differences between rolling a 401(k) versus a 403(b) to IRA while working abroad?
Ideally would like to get the same provider to deliver both these services.
No, there are not. Provided both your 401(k) and 403(b) allow rolling over (in most cases this means you no longer work for the company associated with them), you can roll both into the same IRA.
Where you currently work isn't relevant, unless you still work for either the company that sponsored your 401(k) or the company that sponsored your 403(b).