From Halopedia, the Halo wiki
Containers are a common Forge item. They are available in either open or closed fashion, and are randomly chosen to be red, green, or blue when spawned. When attempting the floating object method, a container can still be moved when physics objects collide with it. Both types of containers cost $6 of budget, and are available on Last Resort and Rat's Nest.
Introduced to Halo 3 in the Heroic Map Pack, boxes function similarly to containers. They come in two sizes; Single Box and Double Box, with both types available either closed or open. Unlike containers, however, boxes only come in their default drab green color. Also unlike containers, boxes are able to properly perform the floating object method. Both sizes of open boxes cost $6 of budget, while both sizes of closed boxes cost only $4. All four box types are available on Foundry, Blackout, Avalanche, and Longshore (though their appearances on Blackout, Avalanche, and Longshore are that of a container), and the open variants of the Single Box and Double Box are obtainable on Ghost Town and Standoff respectively.