They both show the same downward movement on the goodness scale, but with a different starting point. You could translate them both into English as to deteriorate, with some loss of information.
plimalboniĝi - iĝi pli malbona - to get from bad to worse. It already rained, now the wind started and it's even worse.
malpliboniĝi - iĝi malpli bona - to get from good to less good. It's a sunny day, but now the wind started and it's not that nice anymore.
There's a difference between constructions with pli and those with plej.
Plej is a superlative, so we're always taking about one value on a scale, which has to be either the lower or upper extreme.
Pli is a comparative, and talks about two values: the starting point of the comparison, and the point which compares to it.
plimal* and malpli* both situate the compared value lower than the starting point, but imply that the starting point of the comparison was in the lower half, or the upper half, respectively.