Snapchat & Cloud are two different applications within a phone also. I say this just to clear up some here who might not grasp what is being discussed. Good point Shiressleuth..
"Cloud" as we're discussing here isn't an app, as far as I understand.
The concept of "the cloud" refers to the remote storage of data from any app and any device.
Your iPhone saves backups of its essential info (and anything else you tell it to) to Apple's "cloud." Photos also go to Apple's cloud if you choose to back them up in this way.
SnapChat and other media-sharing services post user-created media onto their own "cloud," so that it can be viewed by other users. (In the case of SnapChat, there is a time limit for how long it's available.)
Someone please correct me if I'm wrong.