Etymology
The first modern howitzers were invented in the Netherlands towards the end of the seventeenth century. These were characterized by a shorter trail than other field guns meaning less stability when firing, which reduced the amount of powder that could be used, armies using these had to rely on a greater elevation angle to achieve a given range, which gave a steeper angle of descent
Originally intended for use in siege warfare, they were particularly useful for delivering cast-iron shells filled with gunpowder or incendiary materials into the interior of fortifications. In contrast to contemporary mortars, which were fired at a fixed angle and were entirely dependent upon adjustments to the size of propellant charges in order to vary range, howitzers could be fired at a wide variety of angles. Thus, while howitzer gunnery was more complicated than the technique of employing mortars, the howitzer was an inherently more flexible weapon that could fire its projectiles along a wide variety of trajectories.In other words, there was a marked tendency for howitzers to become more "gun-like" while guns were taking on some of the attributes of howitzers.
In the years after World War I, the tendency of guns and howitzers to acquire each others characteristics led to the renaissance of the concept of the gun-howitzer. This was a product of technical advances such as the French invention of autofrettage just before World War I, which led to stronger and lighter barrels, the use of cut-off gear to control recoil length depending on firing elevation angle, and the invention of muzzle brakes to reduce recoil forces. Like the gun-howitzers of the nineteenth century, those of the twentieth century replaced both guns and howitzers. Thus, the 25-pounder "gun-howitzer" of the British Army replaced both the 18-pounder field gun and the 4.5-inch howitzer. While this had the effect of simplifying such things as organization, training and the supply of ammunition, it created considerable confusion in the realm of nomenclature.
In the US Army, however, the preferred term was "howitzer". What is true for English, moreover, is also true for many other European languages. Thus, as gun-howitzers replaced both guns and howitzers, words such as "obusier" (French) and "haubitze" (German), which had originally been used to designate weapons with relatively short barrels, were applied to weapons with much longer barrels.
Since World War II, most of the artillery pieces adopted by land armies for use as surface-to-surface weapons have been gun-howitzers.

Guns - higher velocity and longer range, single charge propellant, maximum elevation generally less than 35 degrees.
Howitzers - lower velocity and shorter range, multi-charge propellant, maximum elevation close to 45 degrees and upwards. History

A self-propelled howitzer is mounted on a tracked or wheeled motor vehicle. In many cases, it is protected by some sort of armor so that it superficially resembles a tank.
A pack howitzer is a relatively light howitzer that is designed to be easily broken down into several pieces, each of which is small enough to be carried by a mule or a packhorse.
A mountain howitzer is a relatively light howitzer designed for use in mountainous terrain. Most, but not all, mountain howitzers are also pack howitzers.
A siege howitzer is a howitzer that is designed to be fired from a mounting on fixed platform of some sort.
A field howitzer is a howitzer that is mobile enough to accompany a field army on campaign. It is invariably provided with a wheeled carriage of some sort. Examples