They are very different sentences, underneath the surface.
"I will take the children to the airport" is a standard format, not unlike "I will bring the dog into the house." "I will carry the apples to the counter." The "to" phrase is adverbial, and says 'where.'
"Children" [or 'dog', or 'apples'] is a simple direct object.
"take ...swimming" is a sort of double verb situation, not unlike "I will stop him working."
Notice the meaning is different. "take swimming" implies the parent is accompanying the kids to the swimming scene and maybe swimming with them. There are two actions, the accompanying [taking] and the swimming [by at least the children and maybe also the parent]. The location is not stated. Hence you could say, "I will take the children swimming at the athletic club." The "at" phrase [parallel to the 'to' phrase, above] says where.
I agree with Benny and would just add that the definite article is important. "I will take children . . . " would mean children in general, or any children you happen to see. That would clearly not be the case. "The" says that these are particular children.