What is the difference between "normalerweise" and "gewöhnlich"? They both can mean "usually" or "normally". Does it matter which one is used?
"Normalerweise" is obviously related to "normal", which can also be understood as something like "correct". So, "normalerweise" has a bit of a connotation of "this is the way it's normally done, this is the way it should be done".
"Gewöhnlich", on the other hand, is obviously related to "Gewohnheit", meaning "habit", "custom", "practice". So, "gewöhnlich" has more of a connotation of "this is the way we're used to do it, but other ways can be fine as well".
But this difference isn't really big.
To get it out of the way first: gewöhnlich can also be used as an adjective, meaning usual or common, but I don't think you mean that.
Normalerweise und gewöhnlich as adverbs can mostly be used interchangeably. Gewöhnlich is on a slightly more sophisticated language level, normalerweise is more commonly used, you would probably use it more often in everyday conversations.
In more refined language you can use both, and I see hardly any difference. If there's a difference at all, gewöhnlich would probably more commonly be used to just speak about habits and the usual way of doing things:
Wir essen gewöhnlich um 12:30 h zu Mittag.
"Normalerweise" would more often be used when you want to contrast it with a case that is different, like "aber in diesem Fall ...".
Wir essen normalerweise um 12:30 h zu Mittag, aber ich würde mit dieser Arbeit gerne vor dem Essen fertig werden.
It's perfectly valid to use both in both situations though.
The etymology of normal is related to "die Norm", a word that can either mean a common standard or a rule/guideline. In current use, normal and normalerweise refer to a common standard, they are not judgmental per se but only descriptive. In contexts where the common standard is seen in a certain way, they can of course still imply a judgement.
Was unsere Erfindung Ihnen ermöglicht, kann eine Privatperson normalerweise gar nicht selbst machen.
Wenn wir Gäste haben, steht normalerweise mein Mann / meine Frau in der Küche.
As a German native speaker I find the example „Wir essen gewöhnlich um 12:30 h zu Mittag.“ odd or even wrong since I would always say „Wir essen für gewöhnlich um 12:30 h zu Mittag“. You might like to confer https://www.duden.de/rechtschreibung/gewoehnlich.