The biggest difference between these two words is that first should always be used instead of firstly, which should never be used. First is itself an adverb, so adding the -ly ending that turns many words into adverbs is completely unnecessary.
Furthermore, firstly provides a much more stiff tone than that required in most technical writing. Yes, I’ve said many times in this blog that more effective technical writing often takes a more formal tone. But excess in anything often produces undesirable results. You don’t want to provide so much formality that you alienate your audience. Balance is key. Using first strikes that balance very well.
The same reasoning applies in using second rather secondly, using third rather than thirdly, and using last rather than lastly. Adding an -ly ending to words that are already adverbs produces less effective technical writing.
In conclusion, first, you should never use firstly. Second, you should never use secondly. Third, you should never use thirdly. And last, you should never use lastly. Choosing your language precisely will produce more effective technical writing. And that will better represent both you and your brands to your audience.