Heat Map uses gradient coloring from GDI+ for graphical representation of data. The result is a representation of intensity with concentric gradient colors. There is no value other than color (RGB) associated with each cell (pixel) of the image. Heat map has been used in many different fields other than GIS and has been adapted by GIS for representing intensity. en.wikipedia.org/wiki/Heat_map
On the contrary, GRID is fully GIS technique where each cell holds a meaningfull value such as elevation, temperature, atmospheric pressure etc. To create a GRID, typically an interpolation algorithm is used for calculating each cell value based on some sample points. Then some gradient coloring based on the cell values can be used to represent the GRID making it look like a heat map. As opposed to Heat Map, GRID is an actual file with cells and values, not just a graphical represention of points. en.wikipedia.org/wiki/ESRI_GRID
Also, to make you better understand, I suggest you read the article How to use GRID with Map Suite. It uses the Map Suite 2.x technology but it is still relevant to help you get the concept of GRID. gis.thinkgeo.com/Support/Discussion...fault.aspx