Strictly speaking it's not needed for simple apps like that.
However, it's difficult to generalise as we don't know your application or your PICAXE.
The resistor can provide protection.
If, for example, you use a PICAXE with a configurable I/O the resistor could save an embarrassng accident if you set that pin to an output inadvertently and pressed the button (i.e. the resistor can prevent a short-circuit and magic smoke).
Or, if the external source voltage can go higher than PICAXE/V+ it could prevent a nasty current going through your PICAXE.
There are probably other reasons which I can't think of.
I would make a habit of inlcuding a resistor until you are 100% proficient with circuit design and coding - it won't hurt and could save pulling your hair out when smelling that wonderful aroma of burning plastic.