When configured in this manner, the transistors are behaving as voltage controlled current sources, with the collector current being dependant on the base-emitter voltage. A small base current flows, but that can be ignored at this point and is more dependant on the temperature, than collector current: Hfe varies widely.
A diode strapped transistor is just a BJT, with its base connected to the collector, so you're only using the base-emitter diode junction. It's better than an ordinary diode, in this case, because if the transistors have the same part number, they'll pass a very similar current, when given the same base-emitter voltage. The collector currents would match more closely, if the two transistors were on the same piece of silicon, as there's a higher chance they'll share the same characteristics and perhaps more importantly, they'll be at the same temperature.