As a ex-artillery officer, here are the principal differences between the types of platforms.
Mortars are high angle only and are incapable of direct fire. Given an equal caliber, a mortar will have a higher rate of fire than a howitzer or a field gun. Mortars (except for the very largest) can be broken down and man packed or carried by improvised transport (like the bed of a pickup truck). For getting directly behind tall intervening terrain with fire, they are a preferred weapon. Most effective against infantry, limited effectiveness against vehicles, emplacements and buildings. Lethality inceases with caliber but portabillity/mobility decreases.
Howitzers are capable of direct fire, indirect fire and high angle fire. They are either towed or self propelled. Can get really big calibers. Very effective against infantry, limited against vehicles and emplacements. Preferred weapon of choice of you don't have airpower and want to level a position, a building or structure. Biggest variety of ammuntion type - illumination, Smoke - Base Ejecting, Smoke - WP, Cannister (anti-infantry direct fire),HE, ICM, DPICM, and smart munitions and variable time and time fused ammunition.
Field Guns are direct fire weapons and in a pinch, can do low angle indirect fire, limiting their range and usefulness. A anti-tank gun is a example of a specialized field gun, for example. Can get to big calibers like howitzers and are either towed or self propelled. Not a lot of field guns are made anymore due to their limitations as tanks have largely taken over the roles the field guns used to provide.
In general, the larger the caliber, the bigger the lethal zone. The larger the round, the smaller the CEP (Circular Error Probable) footprint - a fancy way of saying that if you want to hit a point target, you get the biggest caliber you can get as the round is more stable in the air and less affect by meterological and has a smaller CEP footprint.
The larger the caliber, the more destructive it is to vehicles and structures and emplacements. Bigger is better.
Call or response times are not weapon dependent. They are determined by the communications capability and doctrines of the C3 systems used by the army in question. Lighter weapons like small mortars can be set up quickly and torn down quickly but once emplaced, once a call for fire goes out, it is the C3 systems, crew training and observer training that determine how fast you see a round on the ground.
Combat Mission games try to simulate artillery systems and capability. Why does it take longer to get a 155mm round on the ground verses a 80mm mortar round base on what I stated above? The delay is to simulate the fact that mortars are closer to the enemy than howitzer systems and to reflect time of flight realities. For example, most of the time, a mortar 1 km from the enemy will tend to have a round on the ground sooner than a 155mm howtizer shooting from 7 km away. And the chain of command / communication issues are simulated as well. A US 155mm is not inherently faster than a Soviet built 152mm yet in game, the US player will get fire for effect well before the Syrian player will. This is doctrine and C3I being simulated in game. So the bigger delay in response time is coded into the game to 'simulate' that.
Hope that answers your questions.