What's the difference between explain and elucidate? I think the difference would be the emotion involved in and/or how something is explained and/or how one think about his/her listeners.
At some point it might help to start learning the etymology of different English words. You can check this in an online etymology dictionary.
For example, the word "telephone" is make up of the prefix "tele-" (from the Greek τῆλε) meaning "far", or "far off", and the root "phone" (from the Greek φωνή) meaning "voice", "sound", or "speaker of". So it makes sense that together they mean, "A device used to send voice far away."
Similarly for "elucidate":
... from Late Latin elucidatus, past participle of elucidare "make light or clear," from assimilated form of ex "out, away" + lucidus "light, bright, clear," figuratively "perspicuous, lucid, clear," from lucere "to shine,"
"Elucidate" is certainly a close synonym of "explain", but with a more specific meaning. It implies that the topic is currently dark and unclear (at least to your audience), but you will bring light to make it easily visible.
Because it's a more educated word, you will want to use it with an audience that can understand and appreciate the meaning. You might also choose to rephrase what you say in more formal or didactic patterns. For example, instead of, "Let me explain how to use this software," you might say:
Let me elucidate the use of this software.
Be careful not to use educated words gratuitously, as this can sound pretentious. In particular, "elucidate" can sound condescending because it carries a subtle nuance that your audience is currently in the dark, meaning ignorant.