It's critical due to the fact that if you don't update, any versions of Tweetbot prior to 2.8.8 or 3.2.2 will stop working and you may need to log into your account again. The reason for the update is a small change with how Tweetbot communicates with Twitter as noted on the Tapbots blog.
A couple weeks ago Twitter's Platform Operations group contacted us and asked us to make a small change to the way Tweetbot communicates with the Twitter API. This change will have no affect on the way Tweetbot works, however two things will occur once this change goes live. All versions of Tweetbot prior to 2.8.8 or 3.2.2 will stop working and you may need to log into your account again. So make sure you are running the latest versions of the app and be prepared to authenticate your account once more. We will be coordinating with Twitter to make the final change on May 29th at 10 A.M. PST. Stay up to date via our @tweetbot account. Thanks for your continued support!
So that's that. It won't change how Tweetbot works in any way that's applicable to you, the user of the app but if you don't update then you'll most certainly have to update later and will have to go through the process of re-logging into your account. The update should be live for most folks right now but if you're not seeing it as of yet, just sit tight. It will be along shortly.