Difference between wait() and sleep()

wait is a BASH built-in command. From man bash:

    wait [n ...]
        Wait  for each specified process and return its termination sta-
        tus.  Each n may be a process ID or a job  specification;  if  a
        job  spec  is  given,  all  processes in that job's pipeline are
        waited for.  If n is not given, all currently active child  pro-
        cesses  are  waited  for,  and  the return status is zero.  If n
        specifies a non-existent process or job, the  return  status  is
        127.   Otherwise,  the  return  status is the exit status of the
        last process or job waited for.

sleep is not a shell built-in command. It is a utility that delays for a specified amount of time.

The sleep command may support waiting in various units of time. GNU coreutils 8.4 man sleep says:

    SYNOPSIS
        sleep NUMBER[SUFFIX]...

    DESCRIPTION
        Pause for NUMBER seconds.  SUFFIX may be ‘s’ for seconds (the default),
        ‘m’ for minutes, ‘h’ for hours or ‘d’ for days.  Unlike most  implemen-
        tations  that require NUMBER be an integer, here NUMBER may be an arbi-
        trary floating point number.  Given two or more  arguments,  pause  for
        the amount of time specified by the sum of their values.

wait waits for a process to finish; sleep sleeps for a certain amount of seconds.