First of all, let's make this clear - there is no difference between hand luggage and other luggage on private jets. Your bags either go in the salon with you or are stored in a storage compartment. If you charter a turboprop or light jet, there is most likely a limit to the size and weight of the bags you can bring with you. So if you are planning a skiing trip, please let us know in advance, so that we can find a jet that will be able to accommodate large bags.
On domestic flights your bags aren't inspected as well, so no standing in the TSA lines! For international flights, you are legally required to pass customs, but private jet passengers usually get special service and privacy.
To summarize, usual aviation restrictions don't apply to private aviation. One more benefit to flying private!