The chief benefits of a diesel engine for offroading are greatly improved fuel economy at lower-than-available power outputs and tremendous low-end torque. The gasoline engine has an advantage in being able to produce more HP for a given weight of engine. 

This is a tradeoff, of course. Situations which call for wheel speed, such as crossing bottomless mud or climbing very loose hills, imply superiority for a gasoline engine. Lots of low-speed travel, cruising at 10 MPH, give a distinct advantage to the diesel. The diesel torque superiority becomes apparent when towing large trailers, or an otherwise heavily laden rig. 

The fuel economy advantage during slow cruising is profound. While diesels at maximum power output consume about 75% as much fuel compared to gasoline engine producing the same power, when both engines are producing 20% of their maximum power, the diesel consumes about 35% as much fuel. Of course, both diesel and gasoline engines vary in their particular consumption of fuel. 

Secondary considerations include the greatly improved safety of diesel compared to gasoline; diesel is hard to light on fire even if you try. 

The most common diesel engine in American 4X4s is the GM 6.2L, installed since 1983 in Chevy and GMC pickups, Blazers, and Suburbans, and most recently offered as the only engine option in the Hummer. This engine has a good reliability record, and has mostly overcome the bad publicity generated by the earlier badly conceived, unreliable GM 5.7L diesel. However, many offroaders, including many Hummer owners consider the 6.2L underpowered for their needs. Ford has used a series of diesel engines >from Navistar (was International). The power has improved over time, and dissatisfaction with available power is mostly limited to those who tow large trailers in hilly country. All of the Navistar and GM engines are V8s, and aftermarket turbochargers can be bought to increase their power, although the $2000+ price tag gives most pause for thought. 

In 1989 Dodge first installed a turbo-diesel in a full-size pickup, although earlier mini-pickups imported from Mitsubishi did use diesels, some turbocharged. The Dodge inline-6 engine is made by Cummins, and clearly set a standard for full-size pickup diesels with power and torque similar to the truck GM 454, while easily delivering 24 MPG on the highway. To handle the 400 ft-lb of torque, Dodge also includes heavier duty drivetrain components with the diesel package, a Dana 60F up front, a Dana 70 in the rear, a large Getrag 5-spd transmission, and the venerable NP205 transfer case. In 1992 GM offered its latest diesel, the turbocharged 6.5L, with power numbers similar to the Cummins. 

In most parts of the world, the majority of 4X4 vehicles are powered by diesel engines, with modern ones being turbocharged. In particular Range Rovers and Land Cruisers sold outside the US are largely diesel powered, along with the cheaper products associated with these companies, such as 4X4 mini-trucks and Discoverys. While most such products are imported into the US, no manufacturers import their diesel engine versions.