Score. It's much easier to use software to generate a sheet music version of a song if the song is recorded using a click.

Manually fix timing errors. You have a grid to compare notes against so it's easy to spot and fix timing errors. This image shows the audio for quarter notes being tapped on a table as it would appear when recorded against a click.
The numbers at the top are the measure numbers (measure 3 is selected). The beats are shown as small white ticks on the black bar between 3 and 4. Note that all of the notes are slightly to the right of the grid making them late. You simply select the offending note (region) and slide it.

There are also some disadvantages:

• Resulting recording can sound mechanical. Subtle increases and decreases in tempo can add life to a recording. Unless tempo changes are programmed into the software, the entire tune will be at the exact same tempo which can be boring to the listener.
• You might pick the wrong tempo. If you are playing without a click and the tempo is wrong, the band will usually gravitate to the correct tempo. If you record against a click, the tempo of the song is nailed down. Even when using a click, it can be helpful to record a bit of the tune without a click to get the right feel then use the recording software to determine what the tempo is: match the click up after the fact. You can then delete the test recording and start real recording at the correct tempo.
• The click in the headphones can bleed into mics. Though the click is often masked by the drums or other instrument, taking it out after the fact can be a pain in the ass. Check for click bleeding before beginning formal recording.

Notes on sliding tracks

• Drum tracks bleed significantly. Always select and slide all of the drum tracks together. Otherwise you end up hearing the bad sound on the other tracks.
• Turn off :snap to grid". When selecting the region to slide, you normally would select in between sounds to avoid introducing clicks and pops into your recording. Most current software does a good job at fixing this but if you make a selection in the middle of a sound, the software will introduce distortion in the form of ramping the audio to eliminate pops and clicks. If it happens in the middle of a snare hit, it will probably ruin the sound. You'll also want Snap To Grid off when sliding the section to get it land where you want it.
• Don't always slide to exactly on the beat. A lot of instruments play behind or ahead of the beat. It's best to move notes to match notes that sound right.

Recording Without a Click

Some musical styles naturally don't work well with a click (jazz, etc.. ) while others such as techno and hip hop do. Many styles such as pop and rock have to be judged on a song by song basis.

Advantages:

• More relaxed sounding. Slower tunes sound more natural when recorded without a click.
• Dynamic buildup. Increases in tempo as a song progresses can dramatically increase the excitement level.
• The tempo adjusts naturally. Sometimes a preprogrammed tempo can be wrong and won't get picked up as a problem until further down the line. When recording with no click, the band has a much better chance of getting in the pocket/groove of a song.
• Less setup. It's usually not that difficult to set up a click but it does mean that all the players need to have headphones and a useable mix with the click loud which can complicate things.
• No click in the headphones to bleed into the mics.

Disadvantages:

• Fixing timing problems is more difficult. When you record with a click, you have beat markers that make it easy to spot late or early notes. With no click, you pretty much have to handle it by ear.
• No easy scoring. Using the recording software to create a printed score of MIDI data is fairly easy but relies on a timing reference which you don't have without a click. There are ways around it but it's generally a lot easier to generate a score when the music is performed along with a click.