Hi there,
Yes, there most certainly is a difference between a crest and a coat of arms. The gent, being a 'consul' of a Germanic State, may well have sported the heraldic badge of Prussia as a sign that he was a Consul of Prussia. Having looked into the records of British grants of arms, I can find none granted in the surname of HENTIG.
But do not be discouraged, there may be a connection somewhere on the Continent, especially if the gent in question was born in one of the Germanic states.
Hopefully, one of the other readers of this message board will be able to assist you with some research on continental heraldry, as I currently do not have the resources that deal with any areas outside those of British armory.
Good Luck.