Vincenzo's link is very good. As long as you understand the reason for performing alignments, the difference is simply that global alignments align against a reference, local alignments without one.
Global alignments are much quicker to perform for large data sets - typically if you double the input sequences a global alignment will take twice as long. For local alignments they usually scale roughly quadratically, so if you double the input sequences they take 4 times longer to complete. While that's alright for a few hundred sequences, once you start getting into next-generation sequencing levels of throughput the run time can become months or years to complete.
The biggest drawback to a global alignment is that it's an extremely laborious process to create the reference alignment, and if it's not accurate then all your alignments will carry the errors of the reference.