"Neo-Pentecostalism" was one of the earliest names given to the cross-denominational "Charismatic movement." Other names included "full gospel" and "Spirit filled" churches.

That whole "third wave" moniker was, in my opinion, an attempt by some Charismatics to distance themselves from what they considered the excesses of the Charismatic movement. So they invented the "third wave" (the first wave being classical Pentecostalism and the second wave being the Charismatic/full gospel/Spirit-filled movement) that was different from the previous two only with respect to an emphasis on modern day prophets, and a de-emphasis on speaking in tongues as the evidence of Spirit baptism.

They all fit under the broad banner of "Charismatic" however, since the term Pentecostal has come to refer to the institutionalization of the movement in a specific family of denominations known as Pentecostals (Assemblies of God, Church of God, etc). One form of "neo-Pentecostalism" found a home in the Church of the Narzarine denomination (doctrinally almost identical to Methodists except for an emphasis on practical holiness and general acceptance of the so-called "second blessing" (baptism in the Spirit, which to Charismatics of almost every stripe is a second work of God upon believers).