The written document is given very high priority, so parties will be held to what is in the document. Both parties sign at the bottom, as a way of signalling their agreement with the terms specified in the document. If conditions are added or subtracted (by crossing out), especially with pre-printed forms, the "customer" (person who didn't write the contract) can initial such modifications, as a way of clearly signalling that they indeed agree to the deletion of such-and-such clause. Since both parties have a copy of the signed agreement, this is not strictly necessary. The potential issue would be that an unscrupulous person could cross out a clause after the contract was signed, and claimed that they aren't bound by that clause. A comparison of the two copies would then reveal that the unscrupulous person was attempting fraud. There is nothing special about handwriting in or crossing out conditions, except that it poses a potential evidentiary problem as to what exactly was agreed to, if for example one party threw away their copy and then maintained that the crossed-out clause had not been crossed out. (So, keep your copy).
In case you are proposing a scenario where one party is unaware of a change, i.e. at the very last minute Smith crosses something out and signs it, and Jones did not see that happen, then both copies would be the same and Jones would be legally bound to what's in the paper. Smith should announce to Jones that a clause was being deleted. We might suppose that there are innocent reasons why Smith made changes without making an announcement to Jones, in which case the parties do not have an agreement. There may be amicable ways to deal with that situation, but push could come to shove, in which case the written form of the document is generally taken to be the most important piece of evidence (though not always the only admissible evidence, unless you're in Colorado, Florida or Wisconsin).