Generally speaking, contract law attempts to ascertain the intention of the parties.
Consequently, it is a well-established rule of contract construction that, all other things being equal, a term specifically drafted in a negotiated contract will take precedence over "standardized" terms in a general contract. The principle is that the parties had the opportunity, at arms-length, to bargain and negotiate for what mattered most to them. Therefore, to the extent that there are any conflicts between two contracts, the negotiated contract will govern.
However, if the negotiated contract is silent on a particular term, and the parties have otherwise entered into a standardized agreement, the court may look to that agreement, for insight into the parties' intentions.
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If the terms of the contract were very clear, and did not otherwise adopt the T&Cs, yes they should override the T&Cs. But without seeing the contract and the T&Cs, it is impossible to say.
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A subsequent agreement normally can revise or change Terms and Conditions, but a real analysis would require a lawyer to carefully examine the contract and the terms and conditions. My colleagues make good points about where there is no "conflict" but rather silence on a point; and about "meeting of the minds". There are other academic questions, but a review of the written materials will almost always be the best, first approach.
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I agree with the others who have cautioned that any attorney would prefer to see the webiste TOS and your contract. However, your post indicates that you and the other party actually signed a separate written agreement. If you have a clear contract covering scope of work and payment terms and conditions, and that contract does not adopt the site TOS, then I would suggest your relationship with the other party is governed only by the 4 corners of your written agreement. I would caution you though to see if you digitally signed any web document or accepted any TOS by clicking a given link on the website such that you agreed that the site TOS would supercede or not be voided by any subsequent written agreement. If so, you may be bound by the TOS. Good luck.