How fat is fat on a vintage Fender neck?
The difference might be small but the feel is definitely noticeable.
What you see at top is a Fender Classic Series '50s Telecaster neck. This is a "fat" neck. What makes it fat is not just the fact it has a "C" and not a "Modern C" contour but rather something else...
...and that something else is the depth, which is sometimes referred to as taper or thickness.
If you pick up any modern Fender American Professional Stratocaster or Fender American Professional Telecaster, both of those guitars have "Modern C" neck contours. The shape is important to how it feels, but so is the 1st fret and 12th fret depth measurements.
Fender "Modern C" has a 1st fret depth a.k.a. neck thickness of 0.82" (20.82mm) and a 12th fret depth of 0.87" (22.09mm). This is true of both the Stratocaster and Telecaster.
The Fender Classic '50s Telecaster neck seen above has a first fret depth of 0.86" (21.84mm) and 12th fret depth of 0.919" (23.34mm).
In simple terms, we're basically talking about a 1mm difference in neck thickness and a different shoulder shape between the modern and the vintage neck. On the back of the neck, the modern is thinner and flatter while the vintage is thicker and rounder.
Does a 1mm thickness and differing shoulder shape really make that much of a difference?
Yes; this is something that can be felt even before you pluck the first note. As soon as you grip the neck, you will feel the difference.