Cartridges, as they were originally introduced, were small ROM* memory chips, which would plug into a memory-expansion bus, so they would map directly into the address space of the CPU. That last bit is very important, as that meant that once the cartridge was in the slot, there was nothing for the game to load - all its assets and code were already in the address space of the CPU!
Cards (MS, SD, etc), in contrast, are just another file medium, with a data-transfer protocol and all. That is, data from them has to be explicitly loaded into the CPU address space - namely in RAM. Cards' great advantage over mechanical file media, particularly optical, comes from seek times - no mechanical parts - no inertia.
So when did ROM cartridge become cards? That I don't know precisely, but I do know the DS carts were already cards, and not ROM cartridges - you had to read their content into the CPU RAM, before you could do anything with that content. The immediate tradeoff from that was that small amounts of directly-mappable memory was substituted for large amounts of memory that had to be loaded chunk by chunk into the CPU address space.
Another thing that cards proliferated was 'flash emulation' - where you get a flash-based device to respond to the data-transfer protocol via a small controller in the middle, so that you could use much larger flash-based media as the physical storage, but the game console would still see those as game cards. That was also possible to do with the ROM cartridges via a small SRAM buffer which would pre-load the desired content from the flash medium, but apparently that was more tricky to implement.
* CD-ROMs, DVDs and BluRay discs are also ROMs, but their access tech supposes a reading mechanisms with a moving head and a disc spinner - something ROM chips don't need.