Is there a difference between cxar; pro tio, ke; and per tio, ke?

This might not be understandable, because English is not my native language...

ĉar
See PMEG §33.6.
Ĉar begins a subordinate clause that gives a reason or motive to the main clause, which could be used on its own.
La tranĉilo tranĉas bone, ĉar ĝi estas akra: The knife cuts well, because it is sharp.

pro ke
While technically correct the convention is that "pro ke" is not used at all, see PMEG §33.2.3. Use instead "pro tio, ke" in cases where the subordinate clause complements the main one, which would otherwise "hang in the air". The subordinate clause gives a reason to the main one.
Tiam ŝi ekploris pro tio, ke ŝi estas tiel malbela: Then she bursted out crying for being so ugly (English uses a non-finite clause here).

per tio, ke
See PMEG §33.2.3 again.
Similar to "pro tio, ke", but in this case the subordinate clause describes a means by which the main one happens or happened.
Li vekiĝis per tio, ke iu lin skuis: (this might not be the most fluent English but...) He woke by having been shaken by someone (again a non-finite clause, this time in past tense).

Difficult to put one's finger on the difference between ĉar and pro tio ke, but perhaps it is something along the lines of 'because/since' and 'on account of'.

Per tio, ke doesn't give a cause of, or reason for, something so much as signalling how the preceding matter is achieved.