PaaS (Platform As A Service) was always trickiest, but one can sum it up as "I, the provider, provide the machine and networking instances already set up for you."
IaaS (Infrastructure as A Service) is simply "Are the machines available for the customer to put whatever they want to on them? Their own OS, DBs, etc?"
IaaS deals with Storage as well - Does it also deal with Databases? I'd say yes.
Would Big Data be Platform As A Service?