As has been said, ANY digital information can be transmitted on ANY satellite. Some frequency bands are, however, more advantageous than others for various types of data. Since we're talking about DirecTV and Dish Network, let's limit our discussion to linear audio/video broadcast.
To date, four different bands have been used for A/V broadcasting: C band, Ku FSS band, Ku DBS band and Ka band (with a fifth, RDBS, coming soon). Of these, C band is the most resistant to rain fade and other atmospheric interference. Howevre, most C band satellite locations are already occupied, and the much lower frequency requires a much larger reflector on the receiving antenna to obtain adequate gain. For these reasons, DTH use of C band has largely fallen out of favor.
Ku is most commonly used for DTH service in North America. Ku FSS and Ku DBS share similar rain fade characteristics, but FSS service is generally powered by transpoinders with anywhere from 70% to 50% of the power of a Ku DBS transponder. To compensate for the reduced power, the dish reflector must be somewhat larger for FSS than for a pure DBS service. In the old days, Primestar was delivered from a Ku FSS satellite and used an oval dish, just under a meter wide.
DirecTV uses Ka for their HD service, and Ka is used in Europe and Asia to carry a wide variety of content. Because of it's higher frequency (and therefore shorter wavelength) Ka is more susceptible to rainfade than Ku. However, the shorter wavelength also means that a dish of a given size will provide more gain for Ka than for Ku, all else being equal. With a modest increase in dish size, Ka can reach 99.9% reliability (the minimum target for DTH service). (Note however that even 99.9% relaibility means that service could be disrupted for up to about 9 hours a year.)
Each of these transmission mediums have advantages and disadvantages (dish size, rain fade sensitivity, availablity of licenses, etc.). None is inherently better or worse than any other. Dish went the all Ku route because they didn't have any available licenses of their own to use, and the capacity that was available was all Ku capacity. The fact that DirecTV had not only the Ka licenses, but also 2 satellites that could use them (one ready to launch and the other far down the construction path), meant that Dish didn't have time to start building any new satellites...they needed the capacity right away. So they licensed some, redeployed some of their existing satellites, and stayed competitive until they could get some new satellites in place.
It should also be noted that Echostar/Dish Network held Ka licenses as well at one point. They lost them for failing to meet the FCC required milestones. DirecTV would likely have lost theirs as well, if it were not for the fact that Hughes had started construction on the Spaceway satellites.