A bond is a "written obligation binding one person to another to perform some specified action or to pay a sum of money", and a recognizance is a "formal acknowledgement of a debt or some other obligation, with sureties and penalties for enforcement".^[1]
Essentially, a bond was a written contract compelling a person to behave well (ensuring they didn't step out of line at all in the event of a rebellion, for example) or to perform a specific task, or else they would lose money specified by their bond. They grew from earning Henry £3,000 a year in 1493 to £35,000 by 1505.^[2]
On the other hand, a recognizance was a recognition that a person owed the crown debt, but did not necessarily force them to pay the sum immediately, or at all. Instead, it was often held against nobility to ensure their compliance.
Sources: