Why is dynamics in music important?

The volume of how notes are played in a rhythmic pattern is called dynamics in music. It is mostly dependent on the rhythm but can also be inserted as a single instance.

Dynamics is also about how the style of how each note is played. Legato prolongs notes so there is no obvious gap in between the rhythmic patterns. Staccato cuts short the sound of a note.

You're both partly right, but ...

Dynamics refers to both the volume (mezzo forte,piano,mezzo piano, forte) and the execution (staccato, legato, etc. as well as the velocity) of the note. When a bandleader tells you to watch the dynamics, in my experience, he/she is referring to volume--generally implying that the volume will change dramatically over the course of the song based on visual queues.

The execution of the notes is more commonly called "attack" (in my experience at least).