The recommendations for creating dedicated layers for power and ground are very sound and are the result of many decades in the fabrication of such boards.
One advantage concerns your basic stripline/transmission line model - the circuit trace(s) with a fixed/constant dielectric and distance above a sold reference plane will have a consistent impedance.
For your simple board this isn't as critical as on a GHz ARM A9 with DDR3 RAM array, but it will still provide great advantage/improvement to how your PCB signals behave.
Having internal power & ground planes provide these consistent references planes for maintaining the trace impedances pretty well.
Without them, your trace impedance will vary by considerable amount and by whatever other copper they happen to be near, and what is on the other PCB side (which is also far away).
Additionally, the internal power and ground planes can be quite close together (e.g. 9 mils) and they form a relatively "tight" coupling which helps to improve our power supply rail regulation across the PCB, and reduce noise.
About the only PCBs that I design that are only 2 layer are purely analog ones - though even a number of them can benefit from dedicated power and ground plane layers.
Not everything HAS to have these layers, but it is very common for any circuit board with a micro to use them in the past ~25 years or so, unless one is making a very crude, inexpensive toy where a few pennies make a difference and functionality isn't that critical.
Show us your basic layout and we can offer more specific advice to help.