Please help.

Correction: I have Finale 2011, not 2007.

Finale 25. I believe that's the only one you can buy, other than PrintMusic 2014.

What is the difference between print music 2014 and Finale 25?

Finale 25 is the full program. PrintMusic is a reduced version, lacking most of the special features that Finale has.

This page will help you see the differences: https://www.finalemusic.com/products/compare-finale-products/ 

Also, they have announced that PrintMusic will no longer be developed.

If you have Finale 2011 (or 2007 for that matter), there's an upgrade path to Finale 25. You can go through your MakeMusic User Account. Your favorite reseller may have the upgrade also.

The Chrome browser has had issues with the Upgrade process but Safari, IE and FireFox are ok. Some have problems with VPNs and Firewalls that are usually solved by temporarily disabling them. 

If you have problems upgrading, use the Submit a request link at the top of this page.

What do you mean about “my favorite reseller”? Since my trial ends today is it urgent that I do the upgrade now?

Some resellers (Musician’s Friend, Academic Superstore, etc.) occasionally offer Finale at a slightly better price. 

You can continue to use the demo version, but you will not be able to,save or print your work. Once you make the purchase, they will send you the necessary numbers to authorize the product. Don’t worry, it won’t magically vanish from the computer!

What MikeR said.

With a reseller, you purchase the upgrade and are then emailed a code with a link that takes you to MakeMusic.com. You then sign into your User Account and download your purchase. It's just a couple more steps than buying directly. 

My question is can I continue to work on my current doc even though I can’t “save” the work. But Finale will save it until I make the purchase?

Finale doesn't save the document. You save the Finale (.musx) file wherever you want it, on your computer. So the status of the program has no affect on the doc.

And as a caution, people who leave their Finale program open for extended period of time while they are away from their computer have (sometimes) found they could not save the resulting file when they returned; so that would not appear to be a solution for you.

>And as a caution, people who leave their Finale program open for extended period of time while they are away from their computer have (sometimes) found they could not save the resulting file when they returned;<

This can affect Mac users too. Disabling App Nap globally (Mac only) and never letting your computer go to sleep may make it less of an issue but there's no promise your work will be there by the time you upgrade so, the sooner you do so, the better.