To put it very simply: Because Norway was less important than France. The main goal for the Allies in Norway was to cause resource scarcity for the Germans. The goal in France was to keep the perhaps most important military force of the alliance (at that point at least) in the fight and to keep the front against the Germans open. Not being pushed off the continent took precedence over pushing Germany out of Norway.
Germany launches Unternehmen Weserübung, the military operation that would see to the invasion and occupation of neutral Denmark and Norway. The government of Denmark, with a land border to Germany and their backs against the ocean, soon makes the emergency decision to capitulate and cooperate in order to prevent damages to Danish cities and people. King Christian X of Denmark capitulated while not in the same pressure that France was in at the time of its armistice; and the subsequent German occupation of Denmark was by the far the lightest of all Nazi occupations of European territories. Danish domestic matters were to be handled independently from Germany and Denmark continued to support domestic anti-German partisans, which would for example lead to the safe evacuation of most Jews in Denmark to neutral Sweden.
Norway meanwhile, with a larger military force, more advantageous terrain, the North Sea inbetween her and Germany and an overwhelming Franco-British naval advantage over the Germans, makes the decision to stand and to fight.
German leadership had designated leadership of the operation to one Nikolaus von Falkenhaust and his 21st Army Corps, consisting of six divisions (69th Infantry, 163rd Infantry, 181st Infantry, 196th Infantry, 214th Infantry and 3rd Mountain).
The Invasion of Norway was also the only major amphibious operation the Germans carried out during World War II - and its experiences would be the main cause that Germany would from that point on prefer to cross the waters with paratroopers rather than with marines, an example being seen in the Battle for Crete during 1941.
The Kriegsmarine supplied the invasion force with two battleships (Scharnhorst and Gneisenau), as well as several heavy cruisers (Admiral Hipper and Blücher), light cruisers (Köln, Königsberg and Karlsruhe) and dozens of destroyers, torpedo boats, motor torpedo boats and minesweepers.
Entry points for the Germans were at Narvik in the north (~2,000 men), as well as Trondheim in the center (~1,800), Oslo in the southeast (~2,000) and Bergen, Kristiansand, Egersund and Stavanger in the southwest, which was the most accessible part of the country from German-occupied Denmark and the German north sea coast. These troops numbered another 3,000 men in the initial invasion thrust.
So overall, the first German deployment to Norway did not consist of more than 10,000 men.
Over the course of April, most of southern Norway falls into German hands, but numerous naval defeats leave the German ground forces cut off from supply. A few months of guerilla combat would eventually have to lead to German surrender in Norway - unless the Germans should be able to pull Polish, British and French forces out of Norway, leaving Norwegians to fend alone...
Now why would the allies abandon the Germans in Norway so quickly? You know how I mentioned 10,000 German soldiers in the Norwegian invasion thrust? Parts of six divisions?
So how many would the Wehrmacht bring into western Europe? 20? 40? 60? No. Try 141. The Germans rolled into Belgium, Luxembourg and the Netherlands (all at that point neutral), with a majority of their 3.4 million soldiers assembled on Germany's western border, and they were hunting down allied troops with 7,000 artillery pieces, 2,500 tanks and 5,500 planes as well. Add to that the looming threat of at least 250,000 initially neutral Italian troops at the alpine border.
This was a menace that the allies could not ignore - they didn't anticipate their defeat, but they certainly had been taught to not underestimate their enemy after Poland's quick demise in the previous September.
They gambled that pulling troops out of Norway might still see Norwegian victory against Falkenhausen's 21st, and that bringing every available man to France would be able to stand against Brauchitsch's and Rundstedt's onslaught.
Wrong on both counts.
Sources & Further Reading:
Churchill, Winston: The Second World War. [six volumes] (1948 - 1953)
Lunde, Henrik O.: Hitler’s preemptive war: The Battle for Norway, 1940. (2009)
Macksey, Kenneth: Guderian. Creator of the Blitzkrieg. (1975)
Manstein, Erich v.: Verlorene Siege. (1955)
Piekalkiewicz, Janusz: Der Zweite Weltkrieg. (1985)
Zentner, Christian: Der Zweite Weltkrieg. (1981)
The Allied withdrawal from Norway, was to put it mildly, motivated by changes in the strategic situation in the rest of Europe. The relative correlation of forces in the theatre further contributed to Allied misgivings about continuing in Norway. The Royal Navy was also concerned about its ability to resupply Allied forces in Norway without taking unsustainable losses.
The main goal of British and French forces in Norway was to deny the Germans access to Swedish iron ore through Narvik. Ore was mined at the Swedish mines at Kirua. In the summer months, it was shipped through the Baltic from Lulea, but in the winter, this route froze over. There were no rail links further south through Sweden, but Kiruna was linked to Narvik, and so it was shipped from there. Swedish iron ore represented a significant portion of German steel production, and as such it was a key target for Allied planners. The success of Allied troops at Narvik would make it appear that this objective was completed. However, this did not account for the key change in the strategic situation that occurred on the 10th May 1940; the German invasion of France. This gave the Germans access to new sources of iron ore, in eastern France and Belgium. As such, removing German access to Swedish ore was no longer important. The situation in France was also seen as far more critical, and a far more useful place for the Allied troops, especially the French, to be fighting in. While the decision to withdraw from Norway was being made, the Germans were in the process of cutting off the BEF, Belgian Army and French First Army off in Northern France and Belgium. This threatened to destroy much of the most effective Allied field forces fighting in France. France was simply the most important theatre the Allies were fighting in. Removing troops from Norway to fight there made sense, especially since it seemed that it was on the brink of being lost completely.
Secondly, while the Allied forces around Narvik were outnumbered German forces there, they were greatly outnumbered by German forces in southern Norway. At their greatest extent, the Allied forces in Norway never exceeded six divisions; four weak Norwegian divisions, four British brigades, and Franco-Polish forces equivalent to a division. Meanwhile, the German forces totalled eleven infantry divisions, plus elements of a parachute division. While many of the German forces were in the south, they were in the process of moving northwards. Should they make a concerted effort against the Allied force in northern Norway, it was clear to Allied leaders that they could not hold. Added to this was the fact that, should France fall, Germany would be free to move additional troops to Norway; should the Allies not evacuate and France fall, their position in Norway would be completely untenable.
Finally, the supplies and reinforcements required by the Allied troops in Norway had to be shipped in from the UK. This was relatively risky, as the Germans had established a number of airbases in southern Norway and Denmark. Aircraft operating from these bases could strike at Allied supply ships and their escorts. During the Norwegian Campaign, the RN lost two destroyers and a sloop to German air raids; more significant were the ships damaged by air attack, which included two battleships, a carrier, five cruisers, eleven destroyers and a sloop. The cruiser Suffolk, sloop Pelican and destroyer Eclipse suffered heavy flooding and were very nearly sunk, while very heavy casualties were suffered aboard the AA cruiser Curacoa. A number of supply ships, such as the Polish liner Chrobry, were also sunk by German bombers. There were other sources of losses; the British cruisers Effingham and Penelope ran aground in Norwegian fjords, with the former having to be scuttled. The supply convoys were also exposed to attacks from U-boats, and to a lesser extent from E-boats. Without a significant commitment by the RAF, these convoys could not be protected. The RAF was loath to release aircraft for the fighting in France or Norway, believing them to be more important for defending British airspace. The Royal Navy was willing to take losses to resupply key positions, as they showed by the heroic efforts made to supply Malta; however, Norway was not such an important strategic location. Meanwhile, the German convoys could not be cut to the same extent. Their routes were protected from Allied air attack by distance. They were only exposed to attack by submarine or surface ship while travelling the short distance between Denmark and Oslo. Allied attempts to cut these saw several losses; Kelly was nearly sunk by an E-boat while operating in the Skaggerak, while two submarines were lost in the same area.
Sources:
The Royal Navy’s Home Fleet in World War II, James P. Levy, Palgrave MacMillan, 2003
The Battle For Norway, Geirr H Harr, Seaforth, 2010