The answer here is very simple. (Become) damp vs wet.
湿る shimeru shows ⺡– “water” falling on the ground and being sucked up by the 日– “sun”. The kanji is often used for atmospheric moisture, but it can refer to anything damp or somewhat wet, as opposed to really soaked.
The main on-reading of 湿る shimeru is shitsu. You want to know this as it appears in a lot of compounds like 湿度 shitsudo – “humidity level” and 湿気 shikke – “humidity, dampness”.
Note that in 湿気 shikke the つtsu of しつshitsu is replaced by っsmall-tsu to become しっけ shikke. This is absolutely regular and happens almost all the time when a つtsu is followed by an unvoiced consonant in making a compound word.
So remember that damp things shimmer. And you can think of damp sheets shimmering on the washing line for both kun and on readings.
濡れる means wet. Like soaking. You see there is both ⺡– “water” and 雨 – “rain” and also a 而 – “rake”. Why a rake? Well, it is that kind of wet that if you just raked the ground the grooves turn into little rivers.
If you need a mnemonic, just remember to keep new rare things out of the rain, or they will get soaked and ruined.
What about the on reading(s)? Well, unlike 湿, 濡 does not have many on-compounds that are much used. Most compound words use the kun-reading. So I really wouldn’t bother about the on-readings at the learning-the-word stage*. This is one of the reasons why blindly learning on-readings from lists is inefficient and wastes a lot of time.
Note that both 湿るshimeru and 濡れるnureru have transitive すsu-versions: 湿すshimesu and 濡らすnurasu. If you know the First Law of Japanese Transitivity, you will find this entirely predictable and know exactly what they mean!
*Note: It isn’t necessary or recommended to learn on-readings when learning kanji organically as words, but these articles are primarily intended for tying together and clarifying the main points of words/kanji you already know. Of course if you learn them for the first time here – ♪bing-bong-BONUS♪