The site does not update changes to the score after replacing a new score on top of the old one

The site most probably is musescore.com, here though you're on musescore.org...

Jojo-Schmitz is correct in saying that the website musescore.com is where your score is stored and where it should be overwritten by a newer version.

But we do need to check a few points if you are using MuseScore 4 on your desktop computer, with the menu options File > Save to cloud ... or File > Publish to musescore.com...

The first time you save or publish a score to musescore.com, the newly-created URL of the score will be stored in the Source field of Project Properties for your local score (on your own computer). But at least on my computer, it's not possible to see the Source field until you move the scroll bar down in order to reveal the last few fields:

On the second and subsequent occasions when you "save to cloud" or publish the score from Mu4, you should always see the extra option enabled as "Replace the existing online score":

1. If you don't see the extra option enabled as "Replace the existing online score", then the URL is very likely missing from the Source field in Score Properties. It's up to you to add the score's URL in that case, before you save to cloud or publish.
2. If If you do see the extra option enabled as "Replace the existing online score", then your online score should definitely be replaced by the newer version. If that doesn't happen, then it is a fault which should be reported to musescore.com : support@musescore.com