Can a full moon appear brighter one month from another?

There are several things that can make one full moon look brighter than another.

If the moon is closer to the earth, it will appear bigger and brighter

If you're in the northern hemisphere, and the moon is above the celestial equator, it's going through less atmosphere, so it will appear brighter.

If you're lucky like we were here, and a storm went through a day or so before the full moon, the air will be clearer, and the moon will appear brighter.

This past full moon, it was relatively close to earth, and in the northern part of it's orbit as it became full, so it probably was brighter than normal.

Well, ..., yes. Factors affecting the brightness of the full moon are:

1) Albedo: The reflective angle of the moon wrt to Earth and Sun Varies slightly each month.

2) Distance: The lunar distance varies from month to month. The closer the moon is the bigger and brighter it appears.

3) Cloud cover ... this is an obvious factor

4) Transparency: The total transparency of the atmosphere from ground to space. It's calculated from the total amount of water vapor in the air.

5) Seeing: Turbulence and temperature differences that affect seeing for all altitudes

Yes it can, the brightness of the moon depends on a number of facts. The phase of the moon is the primary one. Other obvious factors are Earth-Moon and the Sun-Moon distance as well as the atmospheric transparency and extinction Albedo and its variation is the final major factor effecting Lunar brightness.

Earth moon distance varies.When it is near to earth disk size will be larger and so brightness.