I'm not sure fighter aircraft can be included in the category of areas neglected by the Allies, looking at the British Hurricane and in particular Spitfire. Others weren't far behind - the French Air Force had comparable types starting to come into service but was a couple of years too late, the isolationist stance of the US meant its Air Corps was also lagging in 1939 but the delay before its entry into the war allowed it to catch up.

Co-operation with and support of land forces, on the other hand, certainly was neglected in Britain compared to Germany. Not necessarily Close Air Support specifically as it should be noted that "... the bulk of Luftwaffe missions in Poland and France went towards indirect, rather than direct, army support" (Close Air Support: The German, British, and American Experiences 1918-1941, Richard R. Muller) and "... beyond the Sedan crossings, Luftwaffe CAS was sporadic and limited in effectiveness" (The Air War in France, John Buckley). Nonetheless any sort of effectiveness was an improvement over the RAF, which had not been expecting or planning for a continental commitment prior to 1939.

The lessons the RAF drew from the First World War focused more on heavy losses suffered during strafing attacks and the potential of long-range bombing extrapolating from the German airship and bomber attacks on London, and the actions of the Allied Independent Force formed in the closing month of the war. As technology improved and larger bomb loads could be carried more quickly the most extreme fear (or hope) was the "knock-out blow", a brief campaign in which a large number of bombers with incendiary, explosive and gas bombs would cause such devastation to cities that a country would be forced to surrender in days, if not hours. The RAF needed to thus firstly defend Britain against such an attack - initially the only defence seemed to be possessing sufficient offensive capability to act as a deterrent (the basis of Baldwin's famous "The bomber will always get through" speech of 1932), though the introduction of the Hurricane and Spitfire in conjunction with the Chain Home radar network and Dowding's command and control system showed their worth in the Battle of Britain. Secondly, some believed that independent air power alone would be sufficient to bring down Germany (or another continental opponent), though of course that proved to be wildly over-optimistic with the force available in 1945 let alone 1939.

The independent strategic focus was also fuelled by inter-service rivalry. As Muller puts it (perhaps a touch strongly) "... the raison d'etre of the RAF through much of the interwar period was not to prepare for a future war against a real or imagined constellation of hostile powers. It was, rather, to insure its own survival as a separate service." Missions involving close co-operation with the Army risked a loss of control and influence jealously guarded by the Air Force. It therefore drew selective lessons from both the First World War and Spanish Civil War to maintain its strategic focus that persisted throughout the Second World War, though it (somewhat grudgingly) did develop an effective Tactical Air Force starting in North Africa with the Desert Air Force and continuing with the 2nd TAF in Normandy and beyond.

Further reading:

The Paths of Heaven: The Evolution of Airpower Theory, ed. Col Phillip S. Meilinger
Military Innovation in the Interwar Period, ed. Williamson Murray and Allan R. Millett
The Next War in the Air: Civilian Fears of Strategic Bombardment in Britain, 1908-1941, Brett Holman
'A Matter of Faith': British Strategic Air Doctrine before 1939, Malcolm Smith, Journal of Contemporary History

First of all I must disclose that I have only extensively studied the development of the Luftwaffe in the interbellum, and as such can only comment on what I think sets their development apart from other air forces. I will argue that the constraints contained within the Versailles treaty, both with regards to the shape and size of the German armed forces and the constraint on (military aviation, and the Spanish civil war as a testing ground distinguish the development of the German Luftwaffe from the development of other airforces.

In spite of the Versailles treaty General Hans von Seeckt managed to plant the seed for what would become the Luftwaffe. Probably because they could not built openly build an actual full sized air force in the early years following WWI (Resulting from the Versailles treaty) much time was spent by him and his specially selected officers reviewing the performance of their Lüftstreitkrafte during WWI, in which he emphasized the necessity for different branches of the armed forces to cooperate in a form of integrated warfare. He foresaw that maneuver warfare, combined with the use of airpower would be a key to victory in future conflicts, so that a ‘sitzkrieg’ might be prevented.

Now at this time the ‘Strategic Bombing’ theory of Giulio Douhet was very popular amongst airpower theoreticians. In a (unreasonably short) nutshell he theorized that a war can be won by terror bombing against the civilian population of the opponent, they would become so demoralized they would revolt against their own government, thereby forgoing the need for mass casualties amongst the own ground-forces. (Or so he thought).

Now there is a dichotomy between the experiences of the Germans and Allies in terms of strategic bombing during WWI. The allies, in the form of raids by German Zeppelins and later bomber aircraft, suffered far more bombing raids specifically against the civilian population than they were able to direct onto German targets. It sorted some effect and created some panic amongst the civilian London population.My speculation is that this experience contributed to the credibility to Douhets theories in the UK. On the other hand, the bombings raids were very costly to the German’s and for them did not achieve any real strategic effects.

Additionally there is a political argument at play, Tactical use of airpower, or close air support if you will, by its very nature requires close integration with ground and/or naval forces. Strategic bombing not so much. As most airforces were part of the army or navy during WWI, the strategic bombing argument was to plea for an independent airforce. This argument also extended into politics, as u/Bigglesworth_ already mentioned as ‘The bomber will always get through’ trope. This problem was simply a non issue in the immediate post WWI German armed forces called the ‘Reichswehr’. By default, resulting from a clause in the Versailles treaty, all military assets were placed under a single command structure, as argued above, quite the opposite from a political power struggle for independence occurred, Von Seeckt cultivated an open academic culture, and managed to use aircraft (Civilian registered, military operated) in every large military exercise. This necessarily was limited to mainly small sports aircraft, thus focusing more on CAS. Related to this is an economics argument, at this point in time Germany was unable to build and sustain a large fleet of strategic bombers.

As the Nazi regime took power from the early 1930’s onwards the armed forces were developed and split, however the mentality and combined arms doctrine survived. It was put to the test in the Spanish civil war, more or less a guerilla type warfare on the ground. Still lacking a heavy strategic bomber Germany sent a Luftwaffe detachment to Spain in support of Franco, The Condor Legion. They were, amongst other things, tasked with supporting ground troops, As a result CAS was further developed, and some pinch points, such as air-ground radio communications, were sorted out. Furthermore dive bombing was evaluated, and found to be much much more effective, accurate, and by extent economic, than straight level bombing.

A final thought: The only heavy bomber, in range and payload comparable to the four-engined allied machines, operated by the Luftwaffe (He-117) did not enter service until 1942. Coincidentally this is when the western front was more or less settled in terms of Ground operations.

Sources

- James S. Corum, The Luftwaffe: Creating the Operational Air War 1918-1940

- Giulio Douhet, Command of the Air (https://permanent.access.gpo.gov/airforcehistory/www.airforcehistory.hq.af.mil/Publications/fulltext/command_of_the_air.pdf)

- Christopher Locksley, Condor over Spain: The civil war, Combat Experience and the Development of the Luftwaffe Airpower Doctrine. Civil wars 2, no1 (spring 1999): 69-99