Does Re-Projecting ShapeFile only update content of *.prj file?

In the particular case of NAD83 <--> WGS84 and you're working with medium accuracy data, e.g. consumer grade GPS, it's not serious if the only the .prj file is changed, as the two are synonymous at that level.

If you're working with geodetic data accurate to fractions of a centimetre or more the app had better be doing real datum transformation and not just changing the prj file. Also don't forget that there are several different NAD83 definitions (NAD83_HARN, NAD83CSRS, etc), that they are only good for North America, and not uniform coverage at that. See Which Geographic Transformation to use for the Yukon? for a long discussion of the differences of this last point and how to choose, including this illimunating graphic absent from most documentation on NAD83 transformations:

The text in the graphic refers to the names of the various NAD83 transformations available in ArcGIS. Determining what the equivalent terms are in other programs, e.g. PROJ4 and FME, is an exercise left to the reader (feel free to expand in the comments, or better yet, post a new question).

I'm pretty sure that the projecting geoprocessing tool transforms the coordinates of each vertex in your features.

Projection writes an entirely new shapefile, in the output coordinate space -- the new .prj file lets you know what reference system the new coordinates are in. So, if you took a WGS84 shapefile with data like:

latitute, longitude, value
32, -120, off santa rosa island

After projecting to say UTM, the shapefile would contain information like:

x (easting), y (northing), value
216577, 3544370, off santa rosa island

Only through the use of the prj file would you know that the x and y coordinates referred to UTM coordinates. In your specific case, the output coordinates will be very similar due to the similarity of WGS84 and NAD83. Make sure that the software you're using knows how to do datum transformations to perform this operation (its a different kind of transformation from projection).