You do not HAVE to close the street tap, but as the stop cock in your property is your "last line of defence" if anything goes wrong, it is obviously a simple precaution, as you could be faced with an uninteruptable flood if you damage your tap or undo the "wrong" nut!
If the stop cock you are working on is not very accessible, it is also easier.
The gland nut unscrews and the packing can be made with wool and petroleum jelly. The area to pack is self evident when you clean it up.