IMO, it would be better to stick with his .gov email, but in comparison to what Hillary did, this is like a traffic ticket versus capital murder.
Given that no classified data was transmitted by Kushner, yes, there is a fairly wide separation of seriousness, though it's more like "insider trading vs murder".
One of the reasons the rules exist is to (try to) keep all government business on the up and up so that, for instance, Kushner can't contact Michael Scott via private email* to circumvent gov purchasing requirements, give him hints on submitting a winning bid, and then give Dunder Mifflin a $1 million paper contract.
*Certainly this can all be done via phone or lunch meeting (or on the tarmac) as well, but the idea is to try and minimize the events, and archive potential evidence if malfeasance does occur.