What are the differences between the console version and PC release

You can find a detailed overview of the differences between all four difficulties in this thread on the 2K forums.

Here is what that thread says about Normal and Classic:

Normal:

1. Base global panic starts at 0
2. Most minor alien activity raises panic by 1 in left unchecked with some events doing up to 3
3. Your soldiers have no bonus hitpoints
4. Aliens have their AI mostly unshackled
5. Soldiers cost $10

Classic:

1. Base global panic starts at 8 (1 per location)
2. If unchecked alien activity causes at least 1 panic, but will often cause 2 or 3
3. Your soldiers have -1 hitpoint
4. All aliens receive bonuses hit hitpoints, accuracy, critical change and / or damage.
5. Alien AI is unshackled
6. You don't receive officer training school for free. You must unlock and purchase it.
7. There are even more aliens per map
8. Soldiers cost $15

They are the same game.

Unfortunately, Google doesn't allow free apps to turn into paid apps in the Play Store, so after the 24 hours were up, the promo version got pulled from the store and therefore won’t receive any updates. You can still opt for the paid version or use the demo version afterwards if you want to see the updates to the app.

So basically the app you have is the same game but the initial version of it.