Well, incompatible sounds like there would be one standard. It's not.
Well, there is just one standard, the white paper ESRI published many years ago, but there are two problems with that standard that have affected ESRI itself:
1) It is a poorly-written standard that allows too many ambiguities.
2) Over the years, most programmers who have created software that is said to write "shapefiles" have ignored the standard, creating what they call "shapefiles" that require files not described in the white paper to function, and so on.
In other words, business as usual. :-)
The above matters less these days given that a loose consensus seems to have arisen based on what the latest versions of ESRI software consider to be "shapefiles."
See http://manifold.net/doc/mfd9/shp,_shapefiles.htm for the 9 discussion, and http://www.georeference.org/doc/import_drawing_shp_shapefiles.htm for the 8 discussion.
See also http://switchfromshapefile.org/ for a classic non-Manifold rant against shapefiles.
If you look at the lists of various ambiguities and limitations arising from what is written in the original white paper, together with a discussion of extensions such as .prj, usually the interchange problem is one of the items in those lists. Try using lat/lon WGS 84 projection, keep all file names and attribute names short and simple non-numeric names, and use only very simple data in the attributes and usually that will work.