What is the difference between beats and wave packets?

Wave packets are the resultant, or effective, wave due to the super positioning of multiple waves overlapping in space. Beats >> (edit) see Emilio's response below :D. They are the response we hear when two (or more) waves are superimposed on each other. This produces an audible high/low attenuation of the sound.

For example, guitarists use beats to tune their strings by playing the equivalent note on a neighboring string and listening for the beats. When the frequencies of the two strings are similar the beat frequency, which is the difference between the two, is low and becomes easily noticeable. From there, the musician can adjust the string tension until the beats disappear and the frequencies are matched.

Beats result from the interference of two continuous waves of slightly different frequencies. Wave packets are of limited duration and can be described as a combination of many waves with a continuous distribution of frequencies.