Tough question.
Answer: Depends!
I know, cop out, but true!
A resident can purchase a "General Elk License" over the counter. This General tag, no quota, is good for any area in the state that has been designated as a General Area. A certain number (33% of historical sales) "General" tags are reserved for out of state hunters. But there is a catch! The nonresident hunter that gets a General tag is limited to a region of the state, and can't hunt in general areas outside of that region! Also, the nonresident needs to apply for a lottery for his "general" tag, because the demand for them exceeds the 33%! Kind of a high chance lottery!
Now to make it even more convoluted, "special permit areas" are also thrown into the mix! These areas are generally more productive with better quality animals and easier access than the "General tag" areas. These "special permit areas" are limited quota, set by our game commission each year. They still are mandated by law to reserve 33% of the tags for nonresidents, though. These tags become easier for nonresidents to acquire over time, as out of state hunters can build preference points for them, whereas residents are not afforded that luxury!
Take, for instance, elk area 100. This area is mostly within Wyoming's Red Desert. The elk can be accessed by vehicle rather than horses, it is nearly 100% public land, the herd is large and the animals are some of the biggest in the state. A nonresident hunter can build points over the years and can generally draw this permit in 10 or so years. Residents, on the other hand, face yearly drawing odds for this area of less that 3%! I have applied for 25 years and still not drawn this tag!
So, you see, It depends! Do your homework!