Air carrier flight crew on taxi out reported a ground conflict with another aircraft.

Narrative

We pushed off of [the gate] in FLL. Another Company aircraft had pushed off the Gates behind us; and we began our push once their push was complete. We started a single engine and awaited clearance. We were cleared to follow Company and taxi out via Taxiway T. We turned on our Taxi light; began to spool the #2 Engine; and when the aircraft didn't move initially; we gave it just a little more power to get moving. As we started moving; another company aircraft that was sitting at the end of Taxiway B; awaiting clearance into the alley for parking; began their taxi into the alley. While they were angling toward the A-side alley; at that moment of them coming off of Taxiway B and us beginning our taxi; we had to completely stop the aircraft to avoid our right wing hitting their right wing. I am not certain whether they were cleared to their gate or not; but I am certain that we were cleared to taxi. As such; we should not have had to stop the aircraft to avoid a collision.I'd have to check tapes to see if [the other aircraft] missed an assignment or if Ramp Control was unclear in their clearance. But our Taxi light was on; and as such; [the other aircraft] should have clarified before beginning their taxi.