Travelling Club or First will get you into the appropriate lounge regardless of any card you hold. Silver card gets you Club lounge if you're travelling BA, Gold gets you Club or First lounge, regardless of who you're flying with. Also, you can take in a guest with the cards, can't do that if only you have the expensive ticket.
Other benefits are priority waitlisting and, for me the best benefit, ability to use Club (silver) or First (Gold) checkin counters, regardless of what ticket you hold - great for cheapo economy ticket going on holiday, saunter up to the First desk! Regular upgrades on long haul from Club to First was always a nice perk of the Gold card, never guaranteed but I seemed to get lucky quite a lot.
Ascending the tiers isn't as easy as it used to be, Europe and USA trips especially take a loooong time to reach it. Silver is doable that way, but you need a lot of trips to hit Gold, best for that is long haul in the other direction i.e. Far East, Oz etc. Good tip is to have a stopover in Bangkok or Singapore, booking as two separate flights then you accrue points for both legs.