What is the difference between a passing tone and an approach tone? I don't really see any difference.

I believe passing tones are always in stepwise motion, just walking from chord tone, non chord tone, to chord tone. Approach tones can skip and just need to hit a note right before the chord tone.
So, a chord progression just C to Am. The bass could hit a C, then a B, then an A, that would be a passing tone. Or, the bass could hit a C, then a G, then the A. That's an approach tone. Or at least that's what I think the difference is. not 100% sure.

Passing tones are one or more sequential non chord tones that connect two chord tones by a stepwise motion.

Approach tones are apparently something outside "standard" music theory used by guitar players to describe a non chord tone that leads to a chord tone.

In "standard" music theory terminology an approach tone can be a suspension, an appoggiatura, an anticipation, a neighbor or incomplete neighbor, a retardation, an escape tone, or a passing tone. (At least that's what I was taught.)

No, they're not outside of standard theory. Piston would've call them "incomplete neighbor notes."

With an approach tone, we only get the "approach" - as if it appears out of nowhere. It is not dependent or defined by a previous note but it is an embellishment of its note of resolution. It usually resolves stepwise. For example, in the Mozart Sonata in F (K332) that we were looking at, measure 49 has the same melody as measure 41, but it is embellished with approach tones (the D#s).

I believe mentioned the incomplete neighbor along with the other non chord tones.

In regards to not being outside of "standard theory," all of the reference I could find, an "approach tone" could be anyone of the standard designations of a non chord tone, and I could not find any list of standard designations that included the "approach tone" terminology. If you know of a standard reference for harmony or counterpoint or something approaching a standard reference that talks about "approach tones," I'd like to know about it.

From everything that I have been able to peruse, the term "approach tone" is used when the author does not want to take the time to define the "standard" non chord tones. Again, if you've seem something else, I'd like to know about it. Thanks!

Yes, you did mention the term, "incomplete neighbor," along with several other different non-harmonic tones, but you didn't seem to specifically make the connection of it with the term, "approach tone." Personally speaking, I learned the term in college, not in a text, so I can't point you to one that refers to it. As I explained above, it's the same thing as an incomplete neighbor tone (I have a text that places it in the appoggiatura category). "Approach tone" is certainly a more accurate term than "incomplete neighbor" relative to what it actually does. I've never seen a text that uses the term, "approach tone," as a general reference to any non-harmonic tone. But, there are many, many theory texts out there...