Moving the site and changing the site URL

I developed a project in a subfolder and moved my site from this subfolder into the root directory like.

from:
site URL: http://www.project.com/subfolder

to:
site URL: http://www.project.com/

the WP URL of my installation remained:
http://www.project.com/subfolder

I only moved the index.php and the .htaccess files from the subfolder, modified them and put them into the root directory.

everything works well as it should. I can navigate through my site, all links work, all images are fine etc.

my question now…

one of my page URL’s for example should now be:
http://www.project.com/pageurl

and if I navigate myself through my site every URL is showing correctly in the adressbar of the browser.

but in the status bar of the browser at the bottom, I still see the old URL of the site like: http://www.project.com/subfolder/pageurl

anyway if I then click on it, it brings me to the correct page http://www.project.com/pageurl.

Now I really do not know if I should modify the URL’s side wide with a plugin like “Velvet Blues Update URLs” or or it all right like it is?

If I copy the URL by mouse over and give it into the adress bar of the browser, I get 404.

But if I click on it, it brings me to the right page.

Can someone explain to me what this is all about?

Thank You all!