Why did Germany have such an easy time pushing into Belgium and France during the second world war as opposed to the first?

I'm afraid I don't have the knowledge to talk about the First World War, hopefully someone else can give some information on that, however I can talk about why the Germans found it comparatively easy in the Second World War, using a previous comment of mine from here previously, and I'll lift some of the answer from there and add some bits too.

Surprising the Allies

As with Operation Barbarossa, comparable surprise was achieved by the German plan of attack. This was, ironically, partially helped by the fact of Hitler's lack of military knowledge. He wanted to attack France immediately after Poland, even though those in the German Chain of Command knew that Germany was not at all ready for war against a more modernised and capable army such as the French army and the British Expeditionary Force.

The German army was never as motorised as it is portrayed in the media, and those divisions that relied less on horses and were partially motorised had a huge lack of functioning vehicles, even more so after Fall Weiss, the invasion of Poland. The Army simply needed more time to obtain and produce the equipment needed for the Wehrmacht. Eric Brown, a British citizen in Bavaria at the time of the declaration of war, war picked up by the SS and taken to the Swiss border after 3 days. They took all his belongings, but returned his car. When asked why, the SS men replied that they had no spares so the car was worthless to them.

Also, the invasion of Poland happened in September, which meant that winter was coming on, which is never a good time to attack; Hitler wanted to attack immediately, while the generals wanted to wait until spring.

Hitler insisted that OKH produce a plan for the invasion of France through the Low Countries, and so General Halder and his team were put to the job. Halder, at that time connected to the resistance and assassination plot against the Fuhrer, believed that to attack France at this time would be madness, so he aimed to make the plan so self-evidently bad that even Hitler himself would have to postpone the invasion until a better one could be found.

However, the ultimate deception plan came to the fore when on the 10th of January a German aircraft was forced to land in Belgium. This aircraft contained plans for the German attack, and they were captured before they could be burned. In the following weeks, German reconnaissance watched allied troop movements all along the front, and it showed that the Allies had been expecting an attack almost exactly the same as Halder's plans detailed. Another event that helped Halder was the French reaction to German movements along the front; it took the French around two weeks to notice a change in German troop placement and adapt to it.

Halder took this as an opportunity to test out a plan that a German Corps commander, von Manstein, had come up with. It was the plan that would become Fall Gelb, Case Yellow. Armeegruppe B would smash through into Holland and Belgium, while Armeegruppe A (with most of the Panzer forces) would attack through the Ardennes forest in Southern Belgium and Luxembourg, with the aim of crossing the Meuse. Based on the previous reaction times of the French, theoretically German forces would be able to make it through the rough terrain of the Ardennes before the French realised what was going on. This would mean that most of the British and French forces would rush into Belgium and the Netherlands to counter Armeegruppe B's feign attack, while Armeegruppe A coming through the Ardennes would be able to push North-West from the Meuse, surrounding and creating a Kessel - a cauldron (often referred to as a pocket) - around the majority of the British and French forces. It was a huge gamble, but Halder knew Hitler would jump at the suggestion of such a daring plan, and so this was chosen as the main plan for the assault on the West, creating the surprise the Wehrmacht needed.

The Resistance by the Allies

It is well known that the German plan worked, and a huge amount of allied forces were surrounded, culminating in Operation Dynamo, the famous evacuation of British and French forces from the beaches of Dunkirk. Major General Sir Edward Spears (Churchill's representative to France at the time) recalled that 'When Gamelin [The French Army's Commander in Chief] came to England he was asked by Churchill where were his reserves and he said, 'I have none' - it was unbelievable. Churchill was absolutely appalled and began to look for means of going on with the war in North Africa, then in Brittany. Anything, you see, to carry on, not to give way. But the French very soon accepted the idea of defeat, really, as surrender. To them it was rather a conception of the old days of royalty when you just exchanged a couple of provinces, paid a certain number of millions and then called it a day, and started off next time hoping you would be more lucky.' As we know, it would not be so. This explains why the French were unable to do anything to react to the German breakthrough, especially with a decent quantity of men originally on the Front Line as well as the BEF trapped in and around Dunkirk.

However, the French did put up some resistance. For example General de Gaulle performed multiple actions against German Panzer forces. The first occurred when de Gaulle attacked and nearly overran the headquarters of the 1st Panzer Division with the 4th Armoured division, although the swift German reaction with Panzers and Luftwaffe support badly battered de Gaulle's forced, which lacked anti-aircraft guns and fighter cover. A lack of fighter cover also foiled another attack by de Gaulle with 150 tanks a few days later.

However, much of the French army's resistance was token or futile. Panzers often simply drove past French troops on the road and shouted at the them to throw down their weapons and march Eastwards, the French often obliging.

In his book 'The Second World War', Antony Beevor describes the state of General Billotte's First Army Group HQ. 'Billotte's headquarters were in a state of psychological collapse, with many officers in tears. Gort's [Commander of the BEF] chief of staff was so horrified by what he had heard from the British liason officer that he rang the war office in London to warn them that the BEF might have to be evacuated at some point.'

In an interview conducted for the TV series 'The World at War', Captain Andre Beaufre of the French High Command talked about the gravity of the situation in the High Command. 'I must confess that the morale of the French High Command was very quickly broken. In fact the night we happened to know that the front had been broken at Sedan, at that time the feeling was that everything was lost, and...I saw General Georges, who was commanding the north-eastern front, I saw him sobbing and saying, 'There have been some deficiencies.' You always have deficiencies like that and that's nothing new, so we made a plan to restore the sitaution. Then on the Somme in the beginning of June, where we had a certain superiority over the Germans because we were covering their left flank while they were attacking Dunkirk, we had given orders to attack and we achieved nothing. So that was the proof for me that the mentality of the High Command was unable at that time to make an offensive which would restore the situation. From that time on I thought it was lost.'

Captain Beaufre also commented on why the French High Command was so unprepared for this new kind of war: 'It was a period of decay, of very deep decay, probably caused by the excess of the effort during World War One. I think generally speaking we suffered from an illness, which is not peculiar to the French, that of having been victorious and believing that you were very right and very clever.'

French morale was often appalling; before the invasion, French soldiers manning the Maginot line were shocked when asked why they were not shooting at German patrols. 'Then they would shoot back!' they exclaimed. Major General Spears commented on this: 'The Phoney War did a lot of harm to the French soldiers, the Maginot line did a lot of harm, and the whole idea that this was an impregnable fortress and all we had to do was just sit there and wait for the Germans to come and shoot them down.'

By the 15th of June, the armistice was signed and France had surrendered, which meant in just over a month France had fallen. British journalist Gordon Waterford recalled: 'The French government went to Bordeaux. There was terrible confusion because all the refugees landed up there eventually. There was a tremendous battle going on within the government and eventually of course Pétain came to the top and agreed to the German terms.'

Sources:

The Second World War - Antony Beevor

The War in the West Volume 1 - James Holland

The World at War - Richard Holmes [Transcripts of interviews from the Tv series)