Difference Between Contrast Enhancement and Texture Enhancement in Images
Usually Contrast Enhancement is Element / Pixel Wise Operation in Digital Image Processing.  

It means it adjusts the value of a pixel based only in its value.
In order to increase contrast we usually push bright values to be brighter and dark values to be darker (S Shaped Contrast).

When people try to enhance Texture they usually refer to the term Local Contrast.
Local contrast means we want to enhance contrast but using a Neighborhood Operation.
In the sense of contrast, we will push a pixel to be darker even if its value is bright but it is darker than its neighbors. Namely we have the locality sense of the operation.
For instance High Pass will increase local contrast.