In the U. S., Marriage Licenses are certifications by the issuing authority (usually the County
Clerk of
Courts) that the couple are legally authorized to marry. If the parties are under legal age set by law (if any) the
Clerk will expect permission from parent or guardian. But they were also largely money-making devices to get a fee from the couple for the license as well as a fee for recording the date of marriage.
Marriage bonds had a totally different purpose: to assure that there was no impediment to the marriage, certified by the groom (or if underage, guardian or other legal representative such as parent, older brother, uncle, etc.) and a representative of the bride (often called the bride's 'surety' or suretor). Each side had to attest to the issuing authority that they had assets to pay the money penalty specified in the
Bond in the event either party was in default. Default could be a prior marriage not ended by divorce or spouse's death, one of the parties running off with someone else, and other circumstances you can imagine.
So the Marriage License was related to conforming to existing law, while the Marriage
Bond was an agreement between the two parties to the impending marriage.