The CC apps work the same as the previous non-CC apps, in that they install and run on your local computer the same as always. The difference is how the programs are paid for and activated. CC apps require you to log into the Adobe CC server to be authorized, and you pay for them a little each month, rather than a larger amount, once every year or two.
To see what PSE12 does different Google for “Elements 12 What’s New” and “Elements 11 What’s New”.
If you’re trying to decide between Elements 12 and Photoshop CC, then maybe download the Photoshop CC trial and see what it does differently. Whether you can utilize Photoshop for what you do with Elements 10 depends on what you do with Elements.
if I use creative cloud can I save finished projects to my computer or is it just saved to your server
Locally (on your own computer) is normal for most people. Same for CC as for CS6 and earlier versions. Cloud storage on Adobe servers is optional.
Do my files need to be stored in the cloud?
No. Your files remain on your local hard drive. You can also choose to take advantage of the ability to synchronize your local files with your Creative Cloud storage. Even if you choose to sync your files to Creative Cloud, you will always have a local copy of your files.
To clarify, PSE is not part of Creative Cloud, so you would be asking about different programs if you're considering that.
The interface of PSE got a serious makeover in PSE 11, so both PSE 11 and 12 look very different from PSE 10. You might also want to look at this, in addition to the suggestions ssprengel made:
http://www.adobe.com/uk/products/photoshop-elements/buying-guide.html compares versions 8 through 12.
Insanity is hereditary, you get it from your children
If this post or another user's post resolves the original issue, please mark the posts as correct and/or helpful accordingly. This helps other users with similar trouble get answers to their questions quicker. Thanks.