They are different expressions.
As you say, “I don’t have money” means that you have no money at a certain moment or along a period of time.
On the other hand, when you are broke, you are lacking of funds, ruined or impoverished, in an advantageous financial situation, or you have been legally declared financially insolvent. You are bankrupted, as, for instance, Bernard Madoff and his unfortunate clients.
José Manuel Rosón Bravo