Why are there dfferences between Textured and Rendered modes? I have assigned a material with an image texture to this cube (I think) and when viewed in Texture mode I see this 



When viewed in Rendered Mode I see this



This suggests I did not properly assign the material but why is it visible in Texture mode ? I know why I get this purple color but the question is why I see this purple color in Rendered Mode and not in Texture mode (this is why this question is not a duplicate imo)
It appears like you have moved the texture after you brought it into blender. You need to relocate the texture. When blender renders it loads the texture from disk (and is unable to find it causing the purple). The material view is using the copy in memory. When the file is saved and reloaded the material view will purple as well.