The 190/60s will have a diameter of 23.97" while the 185/60s will have a diameter of 24.47". The circumferences will both be pi*diameter but the ratio will be the same -- 23.97/24.45 or about 2% difference. I think this is within the variation you get from one brand to another for the same tire size so probably a non-issue, but having said that some dealers *are* funny about the impact of tire size changes on warranty claims. Not sure where you stand on warranty; car may be just old enough to be outside warranty anyways.
Lower profile tires will also tend to handle a tiny bit better (because the sidewalls don't flex as much) and ride a bit rougher (because the sidewalls don't flex as much
).
The tire guy is probably telling the truth that the circumference is close enough not to matter, but on a family car I would go for the 185/65s over 190/60s even so.
EDIT -- this tire circumference discussion is funny to me, since I drive a vehicle (2003 Jeep TJ Rubicon) which was designed for one tire size then fitted with smaller tires just before launch on the advice of the legal folks, so speedometer and odometer are both off by about 4%.
