I’m going to side-step the debate of whether this is a good or bad thing to do from a branding perspective, but instead use it as an example of what impact it has on your web presence.
Anders was sensible enough to create another twitter account which he used to grab his original name of CityRat59 as soon as he changed the name to AndersSporring - but, this process creates an awful lot of 404s. In fact, it creates a “page not found” for all the URLs to every single tweet you’ve tweeted until the moment you changed name. In Anders’s case, this was thousands.
Twitter basically switches your usename in the URL, with makes no attempt to redirect any of the old URLs with a 301.
All old replies still contain the original profile name, and the original profile name is still linked. So if Anders hadn’t grabbed his old account name and let someone else grab it, suddenly all his old statuses would be linking to someone else - Twitter phishing! Your original name would be hijacked, including any link-juice that the old profile may have had.
The replies are relinked to the new profile; which maintains the thread from a twitter perspective, albeit a little confusing due to the mix of old profile name and new.
Broken lists and favourites
It’s not only old tweets that give “page not found”. All your lists and favourites will also change URL with no redirect. Take Anders’s “Social media club Sweden” list. It’s original URL is still index by Google (at the tme of writing), but of course gives a 404. Google hasn’t been told that the list has a new URL.
Like changing domain
The conclusion? Changing twitter name is very similar to changing domain name; but without any control over redirects. If you have an established Twitter account with a long history of tweets (and perhaps a large number of followers) then you will end up with thousands of broken links and related content that has been indexed by search engines and various twitter-related sites.
My advice would be to think carefully about what Twitter name to adopt when you first join, and then think even more carefully before changing it at any point - Changing Twitter name has a wider reaching impact than you may have initially realised.