That relies on the assumption that the transistor is going into saturation and that the current is being limited at the collector side, hence my original statement about limiting the current if they weren't already. We have been shown no schematic, so if the OP were doing something like, say, limiting the base current to control the LED brightness, then the transistor would not be going into saturation, and the Vbe and gain would come into play. That's not necessarily the recommended way to do that, of course, but I felt the need to cover multiple possible configurations."and may have a higher gain than the 2N2222, so be sure that you are properly limiting the current to the LEDs if you weren't already doing so."
This is a technically false statement.
Both transistors will be turned on fully and it is just the voltage between collector-emitter that has to be taken into account.
The collector-emitter voltage across a Darlington transistor is always more than 1.4v
We don't know the current. Because both transistors are to do the same job (replacement) then I will assume low current. The graph for the TIP120 shows a gain of 475 at 100mA. At lower current, lower gain. This part was designed to run at 2A.and may have a higher gain than the 2N2222