Will heavy tires make a difference for climbing?
There is no way tires are different than other weight when climbing. The saying about rotating mass being equivalent to double non-rotating mass applies only when accelerating and even then you have to take into account that tires are quite lightweight compared to rest of bike and rider.

Rolling resistance difference can be noticeable even when climbing, but Conti 4 seasons are among the worst road tires and Schwalbe Marathon plus is one of the best rolling extremely protected tires, so the difference isn't necessarily that great.