Altium Designer, using signal layers as power planes
While there's a lot of good answers already, I felt I should add some distinctions to "when should you use which, generally" and "when should you move from the ideal, because Altium."

Normally Power Plane layers offer a nice and well defined inherent separation of rules and restrictions. As came up before, it allows programs to assume solid copper in complex analysis. It's also more space efficient when defining large copper areas. In some cases manufacturers can use more reliable negative-defined processes, though these are somewhat outdated and inverting a plot is negligible work anyway.
When you send a plane definition layer the entire supply chain assumes high levels of copper coverage, which is key in getting impedance control right at the least amount of cost. 

However...

There are programs, amongst which Altium, in which you cannot always set vital parameters on your plane layers, that you can set on signal layers and/or for copper pours.

For one example, Altium removes via pads by default on power plane layers, without any option to force it to place them anyway. This is all well and good if you're designing a power-supply or an audio amplifier (a manufacturer that needs them for processing stability will just add a 0.05mm ring anyway). But if you're designing signalling for high MHz and GHz ranges you may actually want full control over all bits of your signal path, including the way vias couple on plane layers. Similarly you cannot easily force the program to pull back a power plane from cut-outs or edges by a configurable distance.

This leads me to the advice, rather than just condoning, to always consider signal layers instead of plane layers in Altium specifically (not as a general argument!), unless you need trace impedance analysis or other plane-tied features.
But if you're working professionally enough to want full analysis of done designs, you are likely to be in a domain of frequencies that someone should arrange a license for a tool like ANSYS or Microwave Studio, which works fine for signal layers as reference plane.