Titanium restore vs. ADB restore. Which one is better after a system update? I backed up my phone (HTC ONE M7, Android 4.4.2) with ADB backup -all -apk (one with -nosystem and once without it). I also backed up by Titanium Free (option system + apps). I also copied the media, download, music etc files into my computer.

Then I wiped my phone, factory reset and updated android to 5.0.2.

What would be the best way to restore the apps and the data that I had backed up?
I am worried that by restoring the apps, some of them might be incompatible with the new android version, or the old system files might overwrite the current system files and cause problems, or the residues of the uninstalled apps from the old system are also restored and make my new installation slow.

what would you recommend for the restore procedure?
I would prefer restore apps using Titanium, for the reason that it gives you total control on which user apps you want to restore.  In your case restoring system apps is not a recommended option since you have upgraded OS. It would lead to a mess up. Hence, while restoring choose the option of restoring just user apps with data. Answer how to restore only user apps is covered here 

As a general guideline, best not to restore system apps after upgrading or changing ROM. After changing ROM also it is better to verify and update only those user apps which are not included as system apps in  new  ROM. You may often come across this situation while flashing Gapps recommended with new  ROM