These two terms are often confused as they both utilize two pairs of speaker cables that are connected to one speaker.
Bi-amping involves the use of two separate amplifiers however, which deliver separate voltage and current to the bass drivers and midrange/tweeters. To do this, the speakers must have two pairs of binding posts that can be physically separated, usually by removing connecting wires or a metal bridge. This will separate the bass section from the midrange/tweeter section. There are definite sonic advantages to bi-amping that we can't go into great detail here, but damping factor increases and intermodulation distortion goes down.
Bi-wiring requires the same physical separation of the bass and midrange/tweeter circuitry as bi-amping, however, you are only using one amplifier, so both pairs of cables are attached to the positive and negative binding posts on the receiver or power amplifier. In some cases manufacturers have supplied additional binding posts on the receiver or amplifier to help simplify this type of hook up. Bi-wiring has more subtle performance advantages than bi-amping, but it is also less expensive to do than bi-amping.