The difference is that the password vault program runs as an application on their computer and there is no access to it from anywhere else. The master password doesn't have to be particularly secure if physical access to the computer itself is secure.
Where others you don't trust have physical access to the computer then yes that master password does have to be secure but provided you have entered that before the person arrives to watch you use the computer you can use the passwords stored there to log into various web sites without the watching person seeing you type a password which is the only instance where having the password field on the screen masked serves any purpose whatever.
Losing a password that is only ever entered on your own computer and never leaves that computer is far less likely than losing one that has to be transmitted over the internet and so the password vault method reduces the chances of your having all sites compromised by making it easier to use different hard to guess passwords for each site on the internet. The actual level of security isn't that significantly different from if you keep the passwords on a written list on your desk (which is the only effective alternative way of using hundreds of different passwords for hundreds of different sites).
The only way a master password on a password vault can be captured without being physically there watching it being typed in is if there is a keylogger installed on the computer and if that applies then all of your passwords are compromised regardless of how you handle them.