I'll be making generalizations here, of course, but I don't quite agree with the idea that WWII was fought with the weapons of WWI. Oh, no doubt, plenty of WWI weapons featured in WWII. Plenty were different, however.

If we split things up into the land, sea, and air "triad", it is easier to break things down.

Air: I start with aviation, because this is the category in which one can quite definitively say that the equipment used in WWII was dramatically different from that used in WWI. The rapid advances made in aviation meant that aircraft that were considered advanced in the mid-30s, like the Polikarpov I-16 or the Mitsubishi A5M, were outdated and outclassed by the start of the war. Even the fairly "outdated" biplanes that served in the war, like the Po-2 or the Swordfish, were designs that came about during the inter-war period. WWI aeroplanes were not used in WWII. The pace in which aviation advanced before and during the war was blistering.

Sea: Here, we have more of a mix. Many ships that were built during WWI saw active service during WWII. Capital ships are expensive, and take time to build. Additionally, capital ship construction was limited by the Washington Naval Treaty. As a result, plenty of navies went to war in WWII with a fair amount of ships from WWI, though these ships were generally modernized, often heavily.

Take Japan, for example. Of twelve battleships possessed by the IJN at it's peak during WWII, eight were from the 1910s (Ise, Hyuuga, Fusou, Yamashiro, Kongou, Hiei, Haruna, and Kirishima). Nagato and Mutsu were both laid down during WWI and completed in 1920, leaving only Yamato and Musashi as Japan's only battleships from the 30s/40s. Once again, it is worth noting that all of those older battleships were heavily modernized by the Japanese before the war.

However, it would be a mistake to think that the majority of ships that served in WWII were of WWI vintage. That certainly wasn't the case. Battleship construction was limited by the Washington Naval Treaty and the sheer expense of building the things, but when it comes to things like destroyers, cruisers, and the all-important carriers, the IJN and other navies were for the most part using ships built during the 20s, 30s, and 40s. And if we're talking about tactics, WWII was very different, due to the impact of carrier aviation. Battleships weren't the most powerful ships in WWII fleets, carriers were - and those carriers were not WWI leftovers.

Land: Here we can see that lots of WWI weapons were still in use, often to great effect. Many armies used bolt-action rifles based on designs that were used during the First World War (and before) - the Lee-Enfield, Karabiner 98k (based on the Gewehr 98), Mosin-Nagant, and M1903 Springfield come to mind - mainly because these guns worked. The same goes for sidearms, like the famous 1911 pistol or the Webley Revolver.

It's easy to see why most armies took a conservative approach to their service arms. Service rifles had to be built in huge quantities, issued to vast amounts of troops, and those soldiers all had to be trained how to use and maintain them. It's a huge infrastructure, and introducing newer, untested designs would put it at risk. America did replace their bolt-action service rifle with the M1 Garand, of course. Most nations stuck with WWI-style bolt-actions throughout WWII. And note: I say "style", since these rifle designs were constantly being updated and modified (quite often with the intent of making them easier to manufacture).

Sticking with small arms, WWII also saw extensive use of submachine guns. Whilst the first submachine gun, the MP 18, was designed during WWI, the famous submachine guns of WWII - MP 38/40, PPSh-41, M3, Sten, etc - were all inter-war or wartime designs. The widespread use of submachine guns (something the Soviets really took to the extreme, outfitting entire units with them), was something new to the Second World War.

Some WWI machine guns saw use in WWII, like the Lewis gun. For the most part, the machine guns used in WWII were newer designs. I'm not too well-versed in artillery, but generally, the pattern seems to hold true: some WWI designs soldiered on (often modified to be more capable), supplemented with plenty interwar and war-time designs.

Tanks, much like aircraft, were an area of rapid technological advancement. A few WWI-era Renault FT tanks might have been used during WWII, but they certainly were not the "majority." Tanks from 1940 outclassed those from 1938, and ones from 1942 outclassed those from 1940, such was the speed of development. Very few interwar designs remained in production throughout the war itself.

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Generally speaking, the "higher" the technology was, the more likely it was to be of more recent vintage. More advanced things like aircraft and ships had to be continuously updated or replaced to remain relevant. The "lower" you go down the line, and the more continuity one can see. A Sopwith Camel would be easy pickings for a WWII fighter aircraft, but a WWI rifle wouldn't really be that significantly outclassed by most of the rifles used in WWII. Thinks like rifles, pistols, or even helmets, have to be produced, issued, and maintained in large quantities, and they are more comparatively simple to things like planes and tanks. When it comes to equipping large amounts of standard soldiers, it's often best to go with weapons that are proven to work, after all.