There's no difference, really: To save games' progress you do it like a normal Gamecube game would, like going to the menu and choosing Save, or reaching a checkpoint, each game does it in it's own way.
You can use memory cards, or activate the option for making a virtual memory card on your HD that will save each game separately.
There's really not much mystery on this subject, although some games don't work well with the virtual memory cards (You can look for a Compatibility list in this very site).
Taken directly from the Nintendont thread:
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Using a real memory card
On wii, if you set the memory card emulation to OFF, it will use the real memory card inserted in slot A.
Attention: Do not save Japanese games on the same memory card than non-Japanese games.
Using the memory card emulation (MCemu)
Warning:
Enabling Memory Card Emulation with a card of 16 MB size might cause crashes in some games.
It is recommended to use 8 MB (1019 blocks) or lower for better compatibility.
To generate and enable memory card emulation, set "MC Emulation" to ON in Nintendont options and choose the size of the memory card to use.
The memory card will be created in /saves/.raw located on the same partition than your game.
If you set "multi" as memory card settings in Nintendont options, it will create a single memory card called ninmem.raw which lets you save multiple games on the same memory card file. Japanese games will have their own card file named ninmemj.raw to prevent corruption and formating message with non Japanese games save.
The memory card emulation is not 100% stable and your memory card can be corrupted.
If you care about your game progress, always make regular backups of your savegame files or of your real memory card using GCMM!
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I personally use real memory cards (I find great to have all my saves on them if I go to a friend's house, for example, and they don't have the problems emulated cards can suffer) and I'm yet to have a problem at all. One for PAL and another for US games.