Learn English – Difference between “shameless” and “unashamed”

Deliberately and on purpose can be used interchangeably to mean "intentionally".

But deliberate (adj) and deliberately (adv) also mean "done with care, methodically."

He made the sandwich very deliberately, carefully covering the bread with a layer of peanut butter and then evenly applying the jelly.

Both informal phrases are used as a phrase of time comparison. The usage differs though.

Way back then:
"Ah, those good ol' days. The 1960s. Sigh. Way back then, the skies were clearer and the people were kind."

Way back when:
"Do you remember the day we first met? Twas that sunny day, way back when you still used to wear those horn-rimmed glasses!"

In the former case, the "then" shows that the time period has been mentioned before, unlike in the case with the latter, where the "when" specifies the time with some incident.