In general, transparency is very important to me so provably fair systems do make a difference in my opinion of places. For the games with systems that are not easily calculated by an end user (such as Blackjack) with a decent grasp on math can only be verified by regulators and that's a whole headache to expect to trust if the casino is entirely online and in a different country for example.

Offering a system that you can verify at any time goes a long way to resolving that worry and gives me confidence in any institution even if I won't necessarily verify every bet.