In Mass Effect 1, each of the classes had a few weapon types that they specialized in and could get better with. Mass Effect 3 has done away with weapon skills. Does this mean that weapons will function the same and do the same damage regardless of what class is using them? Or are there still inherent bonuses for using certain weapon types with certain classes?
There are definite synergies between various classes and certain weapon types. First of all, many powers and abilities only work with specific weapons, or give special bonuses when used with specific weapons. For instance, Vanguards have access to a number of bonuses to Shotgun damage, and Soldiers have access to several Assault Rifle upgrades. Infiltrators have access to lots of Sniper Rifle synergies.
Additionally, only Soldiers and Infiltrators experience the 'time dilation' effect when looking down the scope of a Sniper Rifle - this is invaluable for setting up headshots.
Finally, different classes have different weight allowances. Because some weapon types - notably Assault Rifles and Sniper Rifles - are much heavier than others, trying to carry one with a class with a low weight capacity (like an Adept) will result in a large penalty to your power recharge times. This is an important tradeoff, since these classes essentially live and die by effective use of their powers.
TL;DR: While you aren't required to use certain weapon types, there are definite rewards for specializing in something suited to the combat style of your class.
There is definitely a difference in the way damage is calculated for some weapon and class combinations. You may not be able to see the math on your character page in this version, but if you want to get really nerdy and attach a debugger to the program you'll see it.