Grid, TIF and IMG are three formats of raster files.
In ArcGIS Desktop, apparently there's no difference between them.
What is the difference between these three formats?
Back in the dim and distant past, these formats came about from different proprietary software developers. GRID came from ESRI. IMG came from ERDAS and TIFF came from Aldus (are they still on the go?). So that explains why we have the three of them. There are some differences though:
In GIS we have a massive range of raster formats to choose from and not just these three. They all have differences, some are too subtle to care about within one software package. Other differences are more fundamental. Know your package and who your are going to swap data with - then choose your format... but just be careful about portability.
ArcGIS can read those formats and work with them transparently to the user, however each format supports different features such as pixel bit depth and compression. Following on from @MappaGnosis' answer, see ESRI's technical specifications document for a list of what formats support which features (9.3 version).