Bottom Line Up Front: use the later date unless you have a reason not to.
While the names are different and there are some practical differences, filing an intention to marry, applying for a marriage license, and publishing banns of marriage all serve essentialy the same purpose. They allow the administration (civil or religious) the time needed to make sure the proposed marriage will be legitimate.
In traditional Banns there would be announcements made in the church(es) of both parties for weeks prior to the actual wedding which gave time for people to raise any impediments to the marriage (for example already being married, being too closely related, or having taken a vow of celibacy). Banns are still used (or used as an alternative) in many jurisdictions.
Intentions to marry and applying for a marriage license allow the civil authorities time to ensure you are not already married, are a citizen (or have the appropriate approvals from immigration) - a key difference is that they are not publicized like Banns are.
The later date would b the actual wedding date
As Wayne indicated, there are also instances where two ceremonies would be held, particularly if the families lived great distances apart. in this case it is arguable which date is more appropriate but for consistency the later date makes sense.
