I agree with Jaspar that your first paragraph is prompting for an essay, but is your second paragraph asking for examples, meaning references that discuss the difference?

Here are three:

 1. Richard Skemp's article on <a href="http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf">Relational Understanding and Instrumental Understanding</a>.

 2. The five strands of mathematical proficiency described in <a href="http://www.nap.edu/openbook.php?record_id=9822&page=116">Adding It Up</a>. The first two strands are conceptual understanding and procedural fluency.

 3. Phil Daro's 10-minute video on <a href="http://vimeo.com/59262342">Four Levels of Learning</a>.