I think the problem with these two is that they were charged with "illegal transportation of an unlawfully killed deer". The key being that state is saying the deer were killed illegally in Nebraska and if they had been legally killed there wouldn't have been a charge. I may be wrong, but that is what I take from the charge.
But, even if that is the case and you do manage to legally kill a deer in another state there could be issues with bringing it back into your state. The problem I'm talking about is with the state's attempt to prevent the spread of Chronic Wasting Disease. There are many states that prohibit the importion of out-of-state meat if it still has bones attached (the bones being the part that can carry CWD). If you live in one of those states, you'll have to be sure that all meet is deboned before transporting it .
Late edit: After reading it, I'm even more convinced that this was the case. The charge was brought by the US Attorney on a federal charge in Nebraska. The only way they would do that is if the deer were taken illegally in the first place.