Is there any reason why I should have a contractor get a permit himself instead of getting it on my own?
It is just filing paperwork and maybe drawing up schematics.  In my town they like the home owners getting the permits.  For a panel replacement there isn't much to draw.  You probably have to label what you have and what you are putting in.  Just go down to city hall, tell them what you are doing, and they will tell you what they want from you.  Get the permit, save money, and make sure your electrician signs off on it at the end.

Also if you do the permit yourself you will be sure that the electrician is truly doing what you are paying him to do.  He could submit one with 80% of the work detailed and the inspector will not be looking for the other 20%.

And side note:  It is often easier in smaller towns to bundle permits.  Always ask yourself, "Do I have any other work I need to get done?"

And money saving note:  Permits are often based on the cost of the work done.  If the electrician charges you a lot to get the permit then the permit costs more.