Carls' Jr. and Hardee's Burgers
Hardee's was acquired by CKE Restaurants, owners of Carl's Jr. in 1997, at which point the two chains began to merge their brands. Despite different names and only slightly different menus, the two restaurants are nearly indistinguishable.
Hellmann's and Best Foods Mayonnaise
Although Best Foods acquired the Hellmann's brand in 1932, both brands of mayonnaise retain their original names despite identical branding. But the products aren't entirely the same. Each also retains separate recipes with Hellman's remaining more tangy and Best Foods remaining more sweet.
Edy's and Dreyer's Ice Cream
In 1928 Joseph Edy and William Dreyer entered into a partnership to make and sell ice cream under the name Edy's Grand. Their partnership dissolved in 1947. When William Dreyer Jr. took over in 1953, Edy's Grand became Dreyer's Grand.
But after expanding west, Dreyer's ran into trouble distinguishing themselves from another popular ice cream brand: Breyer's. In order to avoid further hassles, Dreyer's re-adopted the Edy's name for markets west of the Rocky Mountains where Breyer's was already established. Today it's the same ice cream, and nearly the same branding but with a different name depending on your locale.
Know of other products sold under nearly identical branding but different names? Post 'em in the comments! If they're good I'll make more GIFs!
UPDATE! By demand: Ralley's and Checkers
Brownberry, Arnold, and Orowheat Bread