A well-crafted plan would have a single default beneficiary provision that applies across the plan, or at least it would clearly state which provision governs in a given case. If I had to guess, I would say the drafters of the plan you describe were feeling cautious about IRC Section 411(d)(6) at the time of a prior merger, so they copied the full text (or most of it) of the other company's plan wholesale into an appendix, and wrote up something that says "this class of participants is entitled to benefits as provided in the core plan document, except to the extent that the applicable appendix says differently".
As for the substantive difference between "participant's children" and "participant's children, per stirpes", the potential difference lies in how any grandchildren would be treated if their parent who was the participant's child is dead. Suppose that the participant has three children, one of whom is deceased, and all three of which have 2 children apiece. If a benefit is distributed per stirpes, then the deceased child is allocated 1/3 of the benefit, and then that third is divided in two and distributed to each of the deceased participant's children. Note that the other grandchildren still get nothing. If a benefit is distributed to the participant's "surviving children", then each of the two surviving children gets 50%. If the benefit is simply distributed to the participant's "children", then there is some ambiguity as to which of the above results was intended by the drafters (or perhaps some different result entirely).