I think that JP2's support for charismaticism came from his ideology of personalism, where the individual has a personal and therefore subjective 'spiritual' experience.
I recall, when I attended the FSSP indult, that the priest once gave a talk on the charismatics. He said that Catholics are not supposed to ask for or seek out the 'extraordinary gifts' of the Holy Ghost, such as speaking in tongues. If these gifts are given to someone naturally, that's fine. But it isn't even required to be in a state of grace for one to receive these extraordinary gifts (which IMO is quite rare anyway, though the priest didn't say anything about this).
What the charismatics don't seem to get is that we are called upon to grow in holiness first of all, and that those extraordinary gifts (when they are legitimate) aren't meant to help with growing in holiness. And as has been mentioned already on this thread, a Catholic can open themselves up to demonic influences by this. FR. Chad Ripperger (FSSP) has said that he's had to say prayers of exorcism over at least two people who were involved to charismatic renewal.
It reminds me of the new-age movement, where people are also looking for 'spiritual experiences.' It doesn't mean that they are bad or evil - just very much misguided. Also, I think that for most of those folks who think that the Holy Ghost is working through them by speaking in tongues and prophecy, it isn't the Holy Ghost at all, but something subconscious instead.