The other shows the modified contract and says there's nothing wrong. How to determine who's right?
It primarily depends on the jurisdiction and the type of contract.
The scenario you describe is known as the last-shot doctrine. In the US the Uniform Commercial Code "aim[s] to abrogate the criticized common-law mirror image rule and its attendant last-shot doctrine". Superior Boiler Works, Inc. v. R.F. Sanders, Inc, 711 A.2d 628, 636 (1998). The purpose is to preempt "undue advantages derived solely from the fortuitous positions of when a party sent a form", Id., (citation omitted). A number of state legislatures in the US have enacted legislation that largely mimics the model UCC.
The premise that "a critical part has been secretly modified" and went unnoticed to the offeror weakens the validity of altered contract. Trans-Aire Intern. v. Northern Adhesive Co., 882 F.2d 1254, 1260 (1989) points out that material alteration by means of the additional terms "are to be construed as proposals for addition to the contract and will not become part of the contract" (citations and internal quotation marks omitted). Incorporation of material alterations requires the express assent of the offeror/counterofferee, Id. at 1261.
Even if strictly speaking the UCC were inapplicable, the offeree's shadiness in altering a contract already signed by the offeror sounds in breach of the implied covenant of good faith and fair dealing (this covenant is implied in many other jurisdictions besides or beyond the US). That is because the offeror's signature in the contract indicates the offeror is ready to perform the contract as is, and does not expect its terms to be materially modified. In such instances, material alterations that go unnoticed resemble the conditions outlined in Restatement (Second) of Contracts at §153 for a contract to be voidable by the adversely affected party.
The counterofferor ought to ensure the offeror's express acceptance of any material alterations.