Platinum Hits is Microsoft's program to recognize and re-release the most popular games made for its' Xbox and Xbox 360 systems at a reduced price. Box art is changed to signify this fact, though how it is changed differs between the two systems.
On original Xbox, the MSRP is lowered to US$19.99. As far as changes to the box itself, the system bar that runs across the top of the front of the box is white instead of black, and the art itself is shrunk slightly to allow a border to run around it, with the "Platinum Hits" designation between the top of the art and the Xbox bar. See the two images below of Grand Theft Auto: San Andreas for an example; the one on the left is from the original release, and the one on the right is from the Platinum Hits re-release. The color of the case differs as well -- instead of the standard Xbox green, the case is a dark gray-silver.
There are two sub-designations of the Platinum Hits line. Platinum Family Hits showcases titles rated E by the ESRB, and cover art is the same as a standard Platinum Hit except the word "Family" is inserted as if a child scrawled it on. Best Of Platinum Hits is a re-re-release program that takes the most popular of the Platinum Hits, drops them to US$9.99 MSRP, and changes the box art slightly again (this time, the words "Platinum Hits" are in red and the art itself is squished a bit more to allow for a "Best Of" bar to appear above "Platinum Hits"). See the example of MechAssault's Best Of Platinum Hits release below.
Xbox 360 titles are lowered to US$29.99 MSRP when they become Platinum Hits. The case is changed from the standard translucent green to a translucent dark gray. The art itself is shrunk significantly and superimposed over a textured gray-silver background, off center, with the Platinum Hits designation in white and green lettering over the top. The system title bar is not changed. See the example of Need for Speed: Most Wanted below.
The Platinum Family Hits line has been carried over from Xbox, though as of yet, there is no corresponding Best Of Platinum Hits on Xbox 360.