Fighter command was mainly a deffensive force that was orgenised around defending the british isles, the most of its efforts were concentrated on this. In France the british squandrons faced an enemy that fought on its own terms, the Luftwaffe was trained and skilled fighting as an offensive tactical body that supports a rapidly advancing ground force. moreover by the time battle of france started the british squadrons in france consisted of many fighters that were obsolete and inadequate. The famous spitfire wasn't deployed in france and participated in Dunkirk flying from Britain. So the british squadrons couldn't really stop the germans. They did perform admirably however, and caused as much losses to the germans as they would in the most hactic days of the battle of Britain.

During the battle of Britain the fighter command held many advantages, such as the radar stations Chain Home and Chain Home Low that could detect and direct intercepting aircraft towards an incoming raid thus eliminating the need for air patrols; it also had bigger reserves since the british airspace was devided between groups. 11th group bared the blunt of the fighting during the battle, and its battered squadrons could be replaced swiftly with fresh squadrons from groups 10, 12 or 13.

The luftwaffe on the other hand lacked many of the advantages it held in a dinamic campaign, it didn't have the whermacht advancing rapidly and threatening the enemy's airfields, installations and industry, so it had to rely on its bomber force, that lacked a heavy bomber aircraft for this operation. During the battle the german bombers couldn't venture on their own since the british fighters were intercepting them and causing unacceptable losses, so they had to have fighter escort for their time above england, which limited the bomber operating range to those of the fighters, which was quite short and didn't allow for much loitering time, so once engaged, the fighters couldn't continue and defend the bombers for long. This also ment that most of the british aircraft industry could relocate northwards beyond the range of the bombers, so no bomber raid could really effect the fighter production significantly.

And let us not forget that during the opperation the german intelligence asserted time and again that the british airforce was on its last legs, nearing total collapse, in opposition of what the pilots stated daily. This, coupled with hitlers order to attack London, was reflected in the shift of focus towards London on September 7th. During the following week the british airforce pilots could rest up, train fresh pilots and get deliveries of new aircraft. The aircraft industry could actually build a surplus of fighters which helped tremendously.

TL;DR the luftwaffe could operate in france with the dinamic ground campaign, but severely lacking in a completely static campaign in Britain. The brits on the other hand were more prepared to defend the british isles than defending france.

EDIT: as per the rest of the question regarding the technologies and tactics. In general the difference in technology between the Battle of France to the Battle of Britain, the main one is the addition of spitfire 1A that had better performance than the spitfire mark 1. The radar system that was the main technological asset during the battle was developed long before, the first radar mast was raised in 1935.

In tactical regard the brits abandoned their three-ship formation in favor of the finger four formation that they copied from the germans and started firing from way closer than they were instructed. That was done during the battle of Britain though. In the latter stages of the battle the brits dedicated the hurricanes to fight bombers and the spitfire to occupy german fighters.

Fighter Command didn't really fight the Battle of France. The RAF deployed an Air Component of the British Expeditionary Force (Lysanders and Blenheims for Army Co-operation with Hurricanes for escorts/air superiority) to France and a separate Advanced Air Striking Force for offensive operations (primarily 1 Group's obsolescent Battles, along with Blenheims and Hurricanes). The command structure was convoluted, and independent from Fighter Command, who participated but from bases in Britain (or temporary relocation to France).

The French Air Force, on paper, should have been more of a match for the Luftwaffe, but were in the process of expanding and re-equipping, and in a poor political state. Faris R. Kirkland's The French Air Force In 1940: Was It Defeated by the Luftwaffe or by Politics? perhaps slightly over-eggs the potential French strength, but gives a good idea of the difficulties they experienced. Gort, the BEF commander, and the French wanted more fighter squadrons sent to France, but the RAF, particularly Dowding, strongly resisted, keenly aware of the need to preserve home fighter strength if (or as it rapidly turned out when) France fell.

James Holland's The Battle of Britain actually devotes more than half of its 900-odd pages to the lead-up to the Battle of Britain, including the Battle of France, well worth a read.