Mods and assets function somewhat different, so I'll explain them seperately.
Assets are by default turned on when you subscribe to them, so whenever you load up a city (save game or new game, doesn't matter), they will be there. Some might need to be unlocked like default buildings, some might be growables and will only appear in right type of zoning and when the areas has the right attractiveness (depends on the level of the building).
Mods are by default turned off when you subscribe to them, so you have to turn them on in the content manager to use them. They will be applied to your game (new or saved) when you load it. Mods are global meaning if they're on, they're on for all saves and all new games. Some mods add stuff to your city, so the save will need those mods to load correctly in the future. Usually that's stated in the description of the mod.
I hope that clears it up a bit