Translation:Their store is open when my store is closed.
Your grammar is wrong. You should have written, "Their store opens when my store closes, but that is not what this sentence means."
Your sentence means your shop is open until (for instance) 5 o'clock and their shop opens at 5 o'clock.
The Duolingo sentence means (for instance) your shop is open from 9 to 5 and their shop is open from 7 to 11.