Both processes rely on the same basic biology — bacteria consuming dissolved organic matter. What differs entirely is where that biology lives: floating free in a mechanically aerated tank, or fixed in place on a stationary bed of media.
Activated sludge is a suspended-growth process: microorganisms float freely, mixed directly into the wastewater inside a mechanically aerated basin, then get separated back out in a secondary clarifier — with a portion of the settled biomass continuously returned to the basin (return activated sludge, RAS) to keep the population concentrated. A trickling filteris a fixed-film process: the same general kind of microorganisms grow instead as a biofilm attached to a stationary bed of rock or plastic media, while wastewater is sprayed over the top and trickles down through it, and air moves through the bed largely on its own. One spends continuous mechanical energy to keep biomass suspended and oxygenated; the other lets the biomass stay put and lets air move mostly by itself. Neither is simply a better or worse version of the other — they're different tradeoffs between energy, footprint, and operating complexity.