Is there any difference between store apps of Windows 8 for PC and Windows 8 Phone?
Short answer is They are very different from each other.
Though the OS for Windows 8 PC and Windows 8 phones looks like they are the same. They are in fact two different OSs.
As such, the apps that run in PC will not be compatible to phones.
Some apps may have a separate version for the PC and another one for phones that looks and works the same way but they are in fact two different versions of the same app. i.e. Internet Explorer for PC and internet explorer for mobile.
Expecting seamless experience from apps side.
This really depends on what app you're using. Some of your user preference is saved to your Live account (like your bookmarks or saved games for example) but this may or may not be synchronized between your devices.
If you have two Windows 8 PCs you can expect that some of your preference in individual apps will be synchronized between them but when it comes to synchronization between a PC and a phone this cannot be guaranteed.