You simply don't have the correct cookie settings.
in vBulletin Options > Cookies and HTTP Header Options > Path to Save Cookies
you need to set them to use the forum path, not the default of '/' (the path should be an option in the dropdown).
Alternatively, you could give each forum a seperate prefix in config.php ;
PHP Code:
// Prefix that all vBulletin cookies will have
// Keep this short and only use numbers and letters, i.e. 1-9 and a-Z
$config['Misc']['cookieprefix'] = 'bb';
The default is 'bb', just give each forum a unique prefix (you don't have to stick to two letters).