I was very interested in Copernic when it came out, but Google changed everything. I've been working recently with a woman who uses Yahoo for everything, and I was astonished at the totally irrelevant and misleading search results they offer. Some day Google might be replaced (I was a big Alta Vista fan at one time) but they are top dog at the moment. I hear good things about Bing, but haven't spent much time with it.
Likewise, Firefox has come a long way. It's fast and stable and the improvements are driven by user input, as opposed to MS' idea of what users might want. Chrome has also improved, but I prefer FF at the moment.
Windows Explorer, I assume you know, is Windows' file manager. If you keep your images in My Docs/My Pictures (As you should to take advantage of both Windows' and Picasa's features) you'll see that it can be confusing. Pictures from a camera, for instance, are often in Folders bearing the default naming scheme of the camera.Pictures from your Paris vacation might be in a folder named 06102009, and the files may be named DSC12345.jpg, DSC12346.jpg, etc. These files and folders can be renamed and shuffled about using Explorer, or you can take advantage of Picasa's uploading features to name the Folder "My Paris Vacation" when you upload them to your computer. Individual files can be batch-renamed (File/Select All/Rename) or individually re-names to something more useful like Paris1.jpg,Paris2.jpg etc. This is much more useful when searching or browsing.
Picasa puts a different front end on the My Pics folder. Not only do you get the thumbnails, but a whole array of tools to manipulate the images. You can crop, adjust color and contrast, make collages and movies, create virtual folders(Albums), move to other folders and upload to Picasa Web Albums to share and create an online back up. Picasa also has a backup feature that keeps track of what's been previously backed up to make subsequent backups easier.
Other photo organizers do all these things, but Picasa seems to be the best at doing the most common tasks for most people.
Again, a bit of study is absolutely necessary to get the most from this, or any software' and the Picasa help files are very good and well written.