Well, as far as the hen is concerned there isn't really much of a difference.  What it comes down to is what you want and what is available to you. 
 
Most people don't recommend giving a first time broody hen more than 6-8 eggs/chicks to care for.  Mail order eggs can give you a bit of an advantage here as there is no "minimum order" of eggs required to provide enough heat in the packing box for shipping and it's usually not that hard to find someone who will sell you only 6 eggs.  Most hatcheries require a minimum order of chicks that is larger than what most people are comfortable letting a first time hen cover.  Yes, there are hatcheries that will allow a minimum order of as few as 3 chicks...although even then that small a number is largely dependent on whether or not you live close to a large city (My Pet Chicken has a minimum order of 3 chicks, but unless you live next to a large city their minimum order is actually 8).  And shipping so few chicks is significantly more expensive than shipping say 25 chicks. 
 
On the other hand, getting shipped eggs to hatch can be quite the challenge.  There are no guarantees with any eggs that they will hatch after the poor broody hen sits on them for 3 weeks straight.  Even with the most careful packaging and best incubation techniques you can still get a zero hatch from shipped eggs.  If one egg is broken when you get the box, then you know that the rest had some significant turbulence too and may be damaged even if they look good.  Even if no eggs are broken, they still may have been shaken up so badly as to make your hatch rate very low.  Then again, some people can get 100% hatches with shipped eggs.  It's pretty much a matter of luck in how your eggs are handled in transit. 
 
What kind of birds you want your broody to raise can also play a part.  As others have mentioned, you can usually get better quality birds by buying hatching eggs than you can buying mail order chicks.  If your main purpose in having chickens is eggs with the entertainment value on the side, then it may not matter and hatchery birds may fit the bill perfectly.  Quality aside, availability also becomes a bit of an issue.  If you wait for a hen to go broody before ordering chicks, there's a fair chance that the kind of chicks you want may not be available.  Hatcheries fill their orders before they set their eggs, so you usually have to order chicks several weeks in advance before their ship dates.  They usually set more eggs than they have orders for so that they can meet the demand of pullets or in case some of the eggs don't develop, so it may be possible to get live chicks ordered on short notice if you have a hen that decides to be broody, but your selection is going to be significantly decreased.  If you don't much care what kind of chicks you add to your flock, this isn't really a problem.  And has the added bonus that you could buy sexed pullets and not have to play the wait and see game of which chicks are male and which are female.  But if you want a particular breed or color there's a strong chance you may be out of luck ordering chicks instead of hatching eggs.