Which C++ library for ESRI shapefiles to choose?

The OGR Shapefile driver from GDAL/OGR directly uses implementation of Shapelib, so there is no difference actually. If you check the OGR source tree, you will find Shapelib files like shpopen.c and dbfopen.c.

Also, Frank Warmerdam is the author of both, Shapelib and OGR and I can confirm myself that Frank keeps the shpopen.c and dbfopen.c in synch between Shapelib and OGR.

To summary, there is no difference between Shapelib and OGR regarding the implementation of Shapefile format specification.

I've found them both to be ok, but I'd choose the ShapeLib library as ogr is a bit heavy/weird for its purpose.

The shapefile format is very simple; if you only have to access a specific/simple set of shapefiles you could consider reinventing the wheel and write the code to access them yourself. I've done this in an embedded app and it didn't take much more time then using these libs.