A combination of the words "Twitter" and "snowflake." Someone on Twitter that gets really defensive over something that's (overall) trivial. This can also apply to politics, gaming, tv shows, etc. They'd mostly throw out derogatory terms like "racist", "homophobic", "ableist", etc. usually deriving from places like Tumblr.
This person on Twitter is being a twit-flake because someone made a sarcastic joke. So they decided to call the person "racist." Then the other person came out saying it was sarcastic. The twit-flake decided to call that person "ableist" since they didn't include "/s" in the middle of their joke. Even though it was easy to identify as sarcasm even without "/s."