If I am not mistaken, 'по-настоящему' and 'как следует' generally mean 'properly' or 'in the right way.' Is there a difference in usage and meaning between these words?
First of all, Как следует may be not just an idiom for "properly", but also a construction similar to either "as it follows from" or "as it ought to".
On the other hand, По-настоящему literally means "for real", "really". So these words are not always interchangeable.
Подумай как следует --> do proper thinking; think twice
Подумай по-настоящему --> do real thinking (mostly unmeaningful)
"По-настоящему" typically means doing something at full strength, without simplification or imitation of some action for some reason, for example:
боксировать по-настоящему, без скидок на неопытность соперника
"Как следует" (the way it should be done) means doing something properly or thoroughly, as one is supposed to do it. The expression is usually opposed to someone's possible revealing their laziness or is based on a precedent of not doing their best. "По-настоящему" can be used in similar meaning without a hint at a person's imperfection (it's a bit 'politically correct').
На этот раз вымой пол как следует. (more friendly) Вымой сегодня пол, только по-настоящему (= как ты это умеешь).