der Zugriff – access
As you see both ‚Zutritt‘ and Zugriff‘ literally means the same – access or entry. But in case of ‘Zutritt’ it’s more related to entry to any space like an access to a room or entrance to a private property or to any industrial area and so on. There you will see the following signboard:

Unbefugten ist der Zutritt Verboten - Unauthorized access prohibited!
One more example with ‚Zutritt‘:

Zutritt nur für Personal - Staff access only!
While for ‘Zugriff’ – it’s mainly a computer language. Access to any data or to a file. For example you are trying to open or access a file or database but the program denies you any access from reading the file. So you experience the following message:

Zugriff verweigert - access denied (Wenden Sie sich an den Administrator - Please contact the administrator)