I replaced the fuel filter on my car at the weekend with a new Bosch one. the new filter came with 4 new washers. When I took the old filter off I noticed that the old washers had rubber inserts. The new ones didn't but I thought nothing of it as they must be right.

Changed the filter tightened everything up and turned the ignition on. The engine end of the filter started to drip so I tightened it up a bit more. Still leaked. Tried undoing and re-tightening but didn't work. Thought I'd put the old washers on and this worked fine. Started the engine though and the fuel tank end started leaking. Tried tightening but no good. Replaced washers for the old ones and it worked fine.

Do I have the wrong washers or am I doing something wrong with the new ones?

Original seals are dowty washers which are superior to copper washers in this application, they can be reused with no problems as long as the sealing lip isn't torn.
I would guess the wrong size copper washer was supplied.