Unlike an automobile engine, aircraft run at high power settings for very long times. In general the engine is run at maximum power for a few minutes while taking off, then at a slightly reduced power for climb, and then spends the vast majority of its time at a cruise setting, typically 65% to 75% of full power. In contrast a car engine might spend 20% of its time at 65% power while accelerating away from a red light, followed by 80% of its time at 20% power while cruising.
Another difference is that if a car engine fails, you simply pull over to the side of the road. If the same occurs in an aircraft, there is a very high chance that it will crash. Aircraft engines tend to spend more effort on reliability than performance for this reason, and even then it was many years before they had the reliability needed to fly over the Atlantic for instance.
Long duration at high powers, combined with the requirement for high reliability, means the engine must have high engine displacement and be built very tough. Aircraft engines tend to be somewhere between 50% and 100% larger than a similar power car engine for a given power output. They also tend to use the most simple parts and include two sets of anything they can afford to carry, including spark plugs, magnetos, and electrical systems.
Another difference in the aircraft application is that the engine needs to be lifted, powering itself into the air. Light weight for any given power, the so called power to weight ratio is one of the most important features for an aircraft engine, and two smaller light engines will almost always be more attractive than one larger engine in a strictly theoretical sense. However the added complexity of "wiring up" two engines often weighs against its use.
Another difference between cars and aircraft is that the aircraft spends the vast majority of its time travelling at high speed. This allows aircraft engines to be air cooled, as opposed to requiring a radiator, which can lead to lower weight and complexity.
At one time all engine designs were new, and there was no particular difference in design between aircraft and automobile engines. This changed by the start of WWI however, when a particular class of air-cooled rotary engines became popular. These had a short lifespan, but by the 1920s the vast majority of aircraft engines were moving to the similar radial engine design. This combined air-cooled simplicity with large displacements, and were generally the most powerful small engines in the world.
Both the rotary and radial engine have one drawback however, they both have very large frontal areas. As planes increased in speed and demanded better streamlining, designers turned to water-cooled inline engines. Throughout WWII the two designs were generally similar in terms of power and overall performace, but the radials tended to be more reliable. After the war the water-cooled designs rapidly disappeared.
For the smaller application, notably in general aviation, a hybrid design in the form an air-cooled inline, almost always 4 or 6 cylinders horizontally opposed, is most common. These combine small frontal area with air-cooled simplicity, although they required careful installation in order to be effective cooled, notably for the rearmost cylinders.
Throughout most of the history of aircraft engine design, the engines tended to be more advanced than their automobile counterparts. High-strength aluminum alloys were used very early in these engines, decades before it became common in cars. Likewise these engines adopted fuel injection quite early, overhead cams, and a host of other features now common in car engines as well. However the relentless financial power of the automobile industry, combined with the almost wholesale disappearance of the piston engine in airplanes, meant that modern car engines tend to be lighter, more powerful, and, perhaps surpisingly, more reliable.
Over the history of the development of aircraft engines, the Otto cycle, that is, conventional gasoline powered engines, have been by far the most common type. That is not because they are the best, but simply because they were there first.
The diesel engine is one of the designs that makes particularly good sense for an aircraft design. In general diesel engines are more reliable and much better suited to running for long periods of time at medium power settings - this is why they are widely used in trucks for instance. Several attempts to produce diesel aircraft engines were made in the 1930s, but at the time the alloys were not up to the task of handling the much higher compression ratios used in these designs, and they generally had poor power-to-weight ratios and were uncommon for that reason.
Another promising design for aircraft use is the Wankel engine. The Wankel engine produces less power for any given size of engine, about 2/3rds that of a conventional design, but does so for about 1/2 the weight and complexity. In this role the power-to-weight ratio is king, and the Wankel makes particularly good sense. Considerable thinking on such designs started in the post-war era, but at the same time the entire industry felt that jets, often in the form of turboprops, would power everything from the biggest to smallest designs. In the end little work was actually carried out, much to the chagrin of many.
Today the piston-engine aviation market is so small that there is essentially no money for new design work. Almost every engine flying is based on a design from the 1960s, using original materials, tooling, and parts. Whereas modern car engines require no maintenance at all (other than adding fuel and oil) for over 100,000km, aircraft engines are now, paradoxically, rather cranky and unreliable in comparison. Several attempts have been made to introduce newer designs into the market, but invariably they face very difficult problems in the small market.