Hi Enobong,
In simple words the basic difference between Ariba On Premise & On Demand(Cloud) is as follow:
Ariba On Premise is installed in Customer server and all the customization is done by Customer technical team whereas Ariba Cloud solution is present in Private cloud, Ariba team just gives the linking to access the Ariba, your data is secured with them and all the customization is taken care by Ariba technical team.
Regards,
B Aditya