Are regulations covering private jets different from those covering commercial jets?
Technically no, but in the spirit of your question, yes.

All aircraft, including those operated for private and commercial purposes, are governed by 14 CFR 91 - General Operating and Flight Rules.

14 CFR 119 specifies additional regulations that govern certain types of aircraft and operations, including larger private aircraft and commercial operations:


14 CFR 125 covers larger (airline style) private aircraft
14 CFR 121 covers the airlines
14 CFR 135 covers commuter and on demand (charter) commercial operators


So it isn't that private aircraft have relaxed regulations, it is just that they aren't required to comply with the additional regulations that govern the airline and charter operators.

There are many, MANY additional regulations that commercial operators have to comply with, including operational and maintenance regulations.

A few example:  


Pilots flying commercially have strict limits on the time that they are allowed to fly or be on duty.  Pilots of private aircraft don't have any limit, other than not to fly if it would be unsafe.
Commercial pilots are subject to more strict medical requirements.
If an aircraft has to divert due to a mechanical problem, it must be reported to the FAA if being operated commercially.
There are strict limits on how far that a commercially operated aircraft may be from a suitable airport, but there are no such limits for privately operated aircraft.
Passengers must be screened and must be checked against government maintained no-fly lists for commercial flights.
An aircraft used for commercial purposes is required to be inspected more often.
With certain (very limited) exceptions, commercial aircraft are required to be operated by a company (certificate holder) with required management personnel such as the Chief Pilot and Director of Maintenance, while private aircraft can be managed by one person (even the pilot).


See the online eCFR for the specific regulations that fall under each part.