Isn't that a typo of 持たせていただきます?;-)
That's not the difference between the level of politness, but more likely the difference between the meaning or the usage. お持ちいたします indeed can be used as a humble expression of 持つ (e.g. to carry your baggage), but is mostly used as a humble expression of 持ってくる (e.g. to bring coffee to you). Whereas 持たせていただきます is used only for 持つ.