1. To highlight - to light something with a color.
2. To emphasize - to make an accent on something.
Yes, we "highlight" important words (with a highlighter), by underlining them and so on. True, when we "emphasise" our words, we accentuate the way we speak by adding stress and modifying our intonation across a string of words.
There is another meaning for "highlight" which is to focus someone's attention on something. When we "emphasise" something, we tell others that it is really important. As you can see, there can be some similarity in their meaning (getting someone's attention), but there are also cases where it is more appropriate to use one than the other.
1. "We were able to highlight (not emphasise) two critical lapses after a thorough scrutiny of the documents." Here, we want to make something the centre of everyone's interest.
2. "We would like to emphasise the importance of keeping accurate records." Here, we want everyone to remember the great importance of something.
3. Grammarly, a free grammar checking tool, is really useful. With the click of a mouse, it can HIGHLIGHT all the likely grammar errors in my writing. (It just draws your attention to your errors, but it does not stress the importance of your errors.)
4. A speaker from TED Talks gave an interesting presentation to EMPHASISE why being nice to our coworkers is ultimately so important for the growth of a company. (It stresses on the importance of being nice to colleagues.)
I hope this helps.