The Assemblies of God is a Pentecostal denomination. One of several Pentecostal denominations (Church of God, Salvation Army, Church of God in Christ, Church of God of Prophecy, several more). All of them have roots in the "holiness" movement, and in Azusa Street "revivals" that were condemned by just about every other denomination that existed at the time.

One of the distinctions of the Assemblies of God is that they believe that speaking in tongues is the "initial evidence" of Holy Spirit baptism, which they distinguish completely from Holy Spirit indwelling.

The Charismatics, on the other hand, are people in and from non-Pentecostal denominations who believe in Pentecostalism. They can be found in just about every denomination and even many non-Christian sects that claim to be Christian (Roman Catholicism and Mormonism for example). Their beliefs on Holy Spirit baptism and the charismata (gifts of the Spirit) vary widely among Charismatics. The great majority of independent (non-denominational) Charismatic churches are basically Pentecostals who want to avoid denominational affiliation. There are loosely-knit Charismatic denominations (they won't call themselves denominations, but that is what they are) like Calvary Chapels and Vinyard Churches.

-R