Not necessarily. Magic items do not always function the way their prerequisite spells work. For example, a flaming sword does not work like a fireball spell, even though fireball can be used in its creation. So even though the spider climb spell is a prerequisite spell for the Slippers, that in itself does not mean they work exactly like the spell. As melkoriii pointed out above, the spell requires bare hands and feet, but the Slippers obviously do not. When it comes to the issue of icy surfaces, I think it is reasonable to conclude that both the spell and the Slippers interact with the icy surface in the same manner, not because the Slippers have spider climb as a prerequisite, but because they actually do work in essentially the same way with the same ability, namely spider climbing (even though the spell requires hands and feet and the Slippers just feet), and the Slippers say that icy surfaces are no good. YMMV.