Ruby, 3844 characters
->a,b{i=0;i+=1while a[i]==b[i];a[0a[i]&&a[i]==b[i];a[0,i]}
Sample run:
2.1.5 :001 > ->a,b{i=0;i+=1while a[i]==b[i];a[0a[i]&&a[i]==b[i];a[0,i]}["global"["global , "glossary"]
=> "glo"
2.1.5 :002 > ->a,b{i=0;i+=1while a[i]==b[i];a[0a[i]&&a[i]==b[i];a[0,i]}["department" , "depart"]
=> "depart"
2.1.5 :003 > ->a,b{i=0;i+=1while a[i]==b[i];a[0a[i]&&a[i]==b[i];a[0,i]}["glove", "dove"]
=> ""