Unable to boot (kernel panic - reboot cycle) after 10.10.4 upgrade Macbook pro early 2011, after 10.10.4 upgrade does not boot then gives the standard "you have a problem" message, then panic message again etc. I've attached an image below:

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After ⌘Command-R reinstallation of OS the first time it works, but the same cycle after rebooting. 

Applecare service installed fresh OS (erased the disk), it works, bit after I restored the system from backup (I needed the files and programs installed) then same problem.  Then restored from the backup an older version (10.10.3), it works OK, but Apple Store suggests an upgrade, and after the upgrade it does not work again (as it was for the first time after the upgrade). 

Any comments/ideas?
There is an issue with the backup, corrupted files etc. Instead of restoring the backup use Migration Assistant to copy your user files and (optionally) applications. Once that is restored and you can verify the system is working you can work on reinstalling any system modifications that the Migration Assistant did not copy.

If the system becomes unstable after running Migration Assistant and transferring Apps and user data. Scale back what you copy to just user data.

Yeah reinstalling system mods and apps can be a PITA but it sounds like you may not have a choice as your backup is somehow damaged. Without knowing exactly what is wrong with it you will get your best results by restoring as little as possible from your old backup.