There were major similarities, but also major differences. This resulted from the similarities and differences between the threats faced by both the Royal Navy and Britain during the wars. The main constants were the British need to contain hostile threats, the need to support Army operations, and their need to defend British trade and colonies, especially the British lifelines across the Atlantic. However, in the First World War, the threat was mainly confined to the North Sea, while in the Second, it required a worldwide commitment.

In the First World War, the RN's most important threat was the German High Seas Fleet. This was a serious threat, but localised to the North Sea. Hence, the RN could concentrate its heaviest ships to respond to the threat. This was the Grand Fleet, based out of Scapa Flow. As there was no pressing need to respond to threats elsewhere, it could pick and choose its battles to give it the maximum advantage. You see this to a certain extent at Jutland, with Jellicoe avoiding a night battle that put him at a disadvantage. There was a slight threat from the Ottoman Empire, but containing this was mainly the responsibility of the French Navy. The Second World War started similarly to the First, with the British facing down only the Germans. However, in contrast to the First, the German surface fleet was able to operate outside of the North Sea, threatening British shipping in the Atlantic. This meant that the British couldn't concentrate their force to anywhere near the same extent. The entry to the war of the Italians in 1940 and Japanese in 1941 meant that the British had to fight on four fronts - the Arctic, Atlantic, Mediterranean and Pacific. This meant that the RN could never really concentrate a superior force on any one of these fronts as they had in the First. It also meant that they had to accept take more risks - winning a battle could free up more ships for battle elsewhere.

Containing the enemy also meant enforcing blockades, running minelaying operations and submarine campaigns. In both wars, a distant blockade of Germany was carried out. The ships carrying out these blockades were, in both wars, requisitioned merchant ships - armed boarding steamers and auxiliary merchant cruisers. However, in the Second World War, the Royal Navy also had to interdict shipping between Italy and North Africa. This was a job mainly carried out by the warships of Mediterranean Fleet. In both wars, minelaying operations could be offensive or defensive. Offensive minelaying involved laying mines on enemy shipping lanes and outside harbours. Offensive minelaying in the RN was in its infancy during WW1, but 1917 would still see 15,500 mines laid in the Heligoland Bight. In the Second, offensive minelaying would form a major part of British naval strategy. Defensive minelaying (which involves minelaying to protect sea lanes and bases) was far more important in the First World War, as it was one of the few effective ASW weapons available. The RN closed the Dover Straits with a mine barrage, and attempted to do the same with the Otranto Straits in the Mediterranean. There was also a (mainly American led) project to close the northern exits to the North Sea. In the Second, mine barrages would again be laid in the Dover Straits. They would also be laid along the East Coast of the UK, and off the North Coast of Scotland. While these barrages were longer than those used in WW1, they represented less of an expenditure of resources - the WW1 Dover and Otranto barrages were actively patrolled, while the WW2 ones weren't. In the Second World War, the RN carried out active submarine campaigns in all three of its primary theaters. The primary mission of the submarines was to kill merchant ships, with sinking warships a close second. In contrast, the First saw the RN's submarine force supporting the Grand Fleet in the North Sea. It's primary target was the German battleships of the High Seas Fleet. RN subs did, however, carry out an effective anti-shipping campaign in Ottoman waters, especially in the Sea of Marmara.

During both wars, the RN had to support the Army's operations. This included both artillery support and the carrying out of amphibious operations. In WW1, shore bombardment was much more important. The naval attempt at forcing the Dardanelles tested the RN's ability at it, while in the North SEa, the RN carried out a major campaign in support of the northern flank of the Allied lines. Amphibious operations were limited - I've written something about planned amphibious operations in the North Sea here. The only major landing was at Gallipoli. In the Second, the pattern was reversed. Amphibious operations were a major feature of operations from Dunkirk in 1940 to the cancelled Operation Zipper (a landing in Malaysia, cancelled due to the end of the war). The RN became experts at carrying out landings and evacuations. Shore bombardment was limited to solely supporting amphibious operations.

Protecting British trade has always been one of the most important tasks of the RN, and this was no different in the World Wars. Both wars saw threats to British shipping, from both submarines and surface ships. However, there were significant differences here. The surface threat from warships in the First World War was heavily limited - the only German warships abroad at the start of the war were the East Asian Squadron, and the cruiser Konigsberg in East Africa. With the exception of the cruiser Emden, the East Asian Squadron did not attempt a serious commerce raiding campaign, instead choosing to attempt to return to Germany. The regular threat had been neutralised by the end of 1914. Converted commerce raiders did exist, but were usually swiftly neutralised. Instead, the primary threat was submarines. The lack of a convoy system, poor ASW weapons, sensors and tactics, all combined to make British warships near useless at defending against the submarine threat. ASW warfare was instead mostly offensive, with the British hunting U-boats, or attempting to attack them in transit. Q-ships were used to trap submarines. In the Second World War, a new threat was added: aircraft. Convoys in all theatres had to cope with the threat of air attack, which was defended against by additional escorts, land-based aircraft, or by the use of aircraft carriers. Surface ships were more of a threat, with German raiders frequently breaking out into the North Atlantic, while Italian warships menaced British convoys at both battles of Sirte. Much of the British surface fleet in both theatres had to be allocated to countering this threat - either by accompanying convoys, patrolling when raiders were out, or by attacking raiders in port. With the interwar inventions of ASDIC (or sonar) and effective depth charges, and a far better infrastructure for training and developing tactics, the RN was much better placed to defend against submarines. RN warships would no longer spend time hunting U-boats, but would instead be tethered to convoys, protecting against any submarine threats to them. This proved to be a much more effective method for protecting against submarine attack.