Is there a distinction to be made between a German soldier in World War II and a Nazi soldier?

The argument about whether or not these are "German" soldiers of "Nazi" soldiers is one which comes up far too often, for numerous reasons, stemming from mere pedantry to concious efforts to disconnect the Wehrmacht from the Nazi cause in what is known as the "Clean Wehrmacht myth". I would first refer you to /u/Astrogator's fantastic response to a similar question some time ago, and further add my own thoughts on this matter, which echo theirs.

While there is a technical argument to be made that not all members of the Wehrmacht (and once they began drafting later in the war, even the Waffen-SS) were literal members of the NDSAP, it is hard to say how much weight that really ought to hold. The cliche, pithy response I often see to this is that we don't call the members of the US Army the "Democrat soldiers", but of course, the United States is also a liberal democracy, which elects its leaders, and is a far cry from the single-party German state in which party, state, and government were for all intents and purposes one and the same. The Democrats are not synonymous with the state, despite their control of the Presidency, nor, come next Friday, will the Republicans be following the inauguration of the President-elect. So whether or not an individual, given soldier was a true believer in the cause of National Socialism, or if they were a reluctant draftee resigned to their fate, they certainly were fighting for the interests and goals of the Nazi Party in a way that simply isn't true in a liberal democracy, however much the party currently in power might set the agenda.

Overall, the argument about terminology in this regard often seems to be merely academic. When saying someone was a "Nazi soldier", while it can mean that they were 'a Nazi who was a soldier', which, yes, we can't actually know unless he happens to be flashing his party membership card in the photo, it can also mean they were 'a soldier of the Nazis', a statement which it is not particularly easy to disagree with. So in the end, while I would say that someone writing academically should perhaps err on the side of proper terminology, and stick with 'German' instead of 'Nazi' unless they mean "Nazi", as far as casual usage goes, I'm certainly in agreement with Astrogator that the argument against it is weak.

Now, as for your second question, well, as I made mention of, there is what is known as the "Clean Wehrmacht Myth". There are several fantastic responses about this on the sub already, but I would direct you to this one by /u/commiespaceinvader as i'm partial to it. The short summary is that "Yes, they were mostly aware, and institutionally complicit."