A monthly season ticket is valid by National Rail services
only on so-called "permitted routes" between the two stations mentioned and at any intermediate stations. The "ticket type" is shown as "Season". They look like this (but more usually on orange ticket stock):
A Travelcard is valid on NR, Tube and DLR services in the Zones, plus all TfL buses and Trams (if it covers any of Z4-6). Travelcards have the TfL roundel and the word "Travelcard" prominently on them:
A rail-only monthly is £109.50; a Z2-6 Travelcard is £155.60.
If you have a rail-only Season as it seems, then you cannot use it to Feltham as this is not on a permitted route for Hampton Court-Vauxhall. I've not checked but I would imagine the only permitted route is the direct one. The fact that both stations are in the same zone is not relevant as it is not a zonal ticket. You will need a ticket from the last station covered to Staines - in most cases this will be Clapham Junction to Staines; if travelling via Weybridge then from Surbiton (or if in the peak; the Surbiton-Staines Anytime Day Return is cheaper than that from Clapham Junction...).
If you regularly make extra journeys within London not covered by your season, the Travelcard is probably a much better deal. If you get one issued on Oyster then even if travelling into zones not covered you'll automatically be charged only the extra from the edge of your Travelcard to your destination provided that:
- You tap in and out.
- You stay within the Oyster area - check if you're not sure (e.g. Staines would not be valid; you would need to buy a paper ticket from either Feltham or "Boundary Zone 6" before leaving your Zones).