Some historians would say that Pentecostals are from the denominations, churches, or the movement started at Azusa Street, and that Charismatics are from the movement started in the 60's.
In the late 60's and early 70's, a 'charismatics' might be an Episcopalian, Presbyterian, Methodist, Catholic etc. who believed in the baptism in the Holy Spirit and speaking in tongues and other gifts.
Later, though, Charismatic denominations and independant churches came into being that took on the name 'Charismatic' but their teachings were more like Pentecostals in a lot of ways.
In the old days, a lot of Pentecostals would not go to the movie house and dressed a certain way (no jeweltry, etc.) Charismatics came along with their women wearing makeup and some of them even drank wine back then.
But nowadays, the movements have kind of merged in some places. There are still Pentecostals, some of them who dress a certain way. But a lot of the Pentecostals have let go of 'clothes line religion' all together. There still are Charismatic elements in some churches, like the RCC in particular, but in a lot of cases, the Charismatics have moved out of the mainline churches. Plus, many other denominations have toned down resistance to spiritual gifts and have at least a theoretical acceptance of it.
The Charismatic movement also includes the word of faith movement. Not all Charismatics are into this. Word of faithers believe in 'name it and claim it.' Some of them emphasize using faith to get money, 'planting seed' by giving and harvesting money, and things like that. These are some of the Charismatics who get the most airtime on TV.
Pentecostals generally baptize by immersion. Many independant Charismatic churches do. (Actually, I don't think I've ever heard of any that sprinkle babies.)
There are also cultural issues. Sometimes people think of Pentecostal churches as having a certain type of lively old timey southern style church music. Charismatic churches tend to have choruses.
But nowadays a lot of Pentecostal churches will have choruses mixed with older hymns.
The term 'charismatic' can also be used as a theological term to describe someone who believes that spiritual gifts continue. Literally, the term would mean someone who believes in or is characterized by gifts (and the word for gifts is related to the word for 'grace.') So literally we should all seek to be 'charismatic.'
There is also a movement known as 'Third Wave' meaning churches of people from evangelical backgrounds that did not accept the Charismatic movement in the 60's and early 70's, but later started to experience spiritual gifts. This is some term a theologian or historian came up with.
Pentecostals and Charismatics believe in speaking in tongues and other gifts. In some of these churches, everyone prays in tongues at the same time. Other churches like this expect there to be an interpretation if there is a tongue. Sometimes, during a lull between songs, or at some other time during the meeting, someone will speak out a message in tongues. In Pentecostal churches the music generally stops and everyone acts reverent just like if someone is leading in prayer. Then, they wait. There may then be an interpretation in tongues. Prophecies work more or less the same way in a lot of churches, but just with one person speaking.