There are a lot of good answers to this question already but another way to take a look at it is the cloud (ala Amazon's AWS) is good for interactive use cases and the grid (ala High Performance Computing) is good for batch use cases.
Cloud is interactive in that you can get resources on demand via self service. The code you run on VMs in the cloud, such as the Apache web server, can server clients interactively.
Grid is batch in that you submit jobs to a job queue after obtaining the credentials from some HPC authority to do so. The code you run on the grid waits in that queue until there are sufficient resources to execute it.
There are good use cases for both styles of computing.
