Generally, it is best to post in one forum only. The reason is that someone may take time to respond but the answer may have already been provided elsewhere. I'll answer the question here and you can post to your thread on the Flash forum that you have received a reply.
The difference is:
Flash Player - plays content with the extension SWF, most often created with Adobe Flash Pro. There are other tools that can publish to swf. Illustrator can even export to swf.
Shockwave Player - plays content with the extension DCR - created with Adobe Director.
The term Shockwave has confused people in the past. Originally Flash was called Shockwave Flash. Now, Flash and Shockwave are separate things. It should not harm your computer to have both installed. They are both trusted web plugins. If you installed them from the Adobe website, you should be fine.
Here are some examples of different content:
Hope that clears things up for you.
hello, dean --
thank you so much for your quick reply and the explanation. i kept going thru the adobe website and reading and reading, but there were so many references to other things i didn't know about -- and they all referred to each other -- that i was getting quite frustrated!
i appreciate your willingness to help. (i will put a cross-reference in my duplicate post to send people to this answer. thanks for the info on that too.)
-- susie margaret
Am happy to help. I understand it can be confusing. Pat's reply in the Flash forum is the more Adobe explanation with a marketing perspective. While Flash and Shockwave are distinct products, there is an overlap and people do confuse one over the other.
Enjoy the content that you can now see with the Flash and Shockwave players.