A cookie as set by the server can be a session cookie, that is meant to be kept around only for as long as the session lasts. curl has no idea how long a session is or when it ends.
Instead of telling curl when a session ends, curl features an option that lets the user decide when a new session begins.
A new cookie session means that all the old session cookies are be thrown away. It is the equivalent of closing a browser and starting it up again.
Tell curl a new cookie session starts by using -j, --junk-session-cookies:
curl -j -b cookies.txt http://example.com/