There are a few key differences between Base, Silver and Gold cards, and those differences relate to cosmetic/grade, ownership/tradability, and rarity.
Re: Grade: Base, Silver and Gold each represent a different cosmetic grade of card. I.e. you could have a Base version of the card Starfield, you could own a Silver version of the card Starfield, and you could own a Gold version of the card Starfield. Base, Silver, and Gold cards are visually distinct from one another. The cosmetic differences provide no competitive gameplay advantage.
Re: Tradability: Base cards cannot be sold, purchased or traded. Silver and Gold cards, on the other hand, can be sold, purchased and traded.
Re: Rarity: Base cards are unlimited in supply and are unlocked via accumulating XP. Silver cards are created by players through Ranked Rewards (i.e. ranking highly on the leaderboard) and Conquest. The supply of Silvers is controlled by the number of players in the game and the number of Conquests played. Therefore, Silvers are rarer than Base cards. Gold cards are created by players who win all three matches in a given Conquest. The supply of Golds is determined by the number of Conquests won by players. Golds are rarer than Silvers.