Why did the Germans build battle cruisers with a different profile than the British?

British favored firepower and speed. IN any battleship (or battlecruiser) you negotiate the Triad. A compromise between three variables

1. Speed (horsepower)
2. Firepower (offense)
3. Armor (Weight)

A ship with smaller guns carries more ammunition......but less hitting power from ranges past 10,000 yards. A ship with More armor, weighs more thus engine power is insufficient to keep up with faster cruisers and destroyers. A ship with better engines is faster.

German's stressed armor because they had a smaller navy and couldn't afford a more robust fleet. Britain lost more ships during the Battle of Jutland.........twice as many tonnage

The reason appears to be a change in German thinking in the use of warships.

In the earlier phase, up to about 1930, the purpose of German warships appeared to be as "duel" ships. In that context, it made sense for the Germans to build more heavily gunned ships than British ships of equivalent speeds; that is, pocket battleships to fight cruisers, and later, superbattleships like the Bismarck to fight British battleships, etc. The German battlecruisers Scharnhorst and Gneisenau did not fit this mold, which appears to be the basis of your question.

There appears to have been a change in the early 1930s, when the Scharnhorst and Gneisenau were built as commerce raiders. Their speed was clearly an advantage, but the German navy determined that the longer-ranged11 inch gun was optimal for commerce raiding compared to shorter range but heavier 15 inch guns. So they made their "battlecruisers" supersized versions of their pocket battlehips, rather than lightly armored "battleships" with 15 inch guns for "duels" against British battlecruisers.

Later, when Hitler took power (and asserted his control over the Navy), it went back to building larger "duel" ships such as the Bismarack and Tirpitz.

Renown was a true battlecruiser in the Jackie Fisher sense. It was built during World War 1, and intended to fight other similarly armed battleships and battlecruisers of the German high seas fleet. Post Dreadnought both Germany and the UK built large fleets of such ships, with the distinction between battleship and battlcruiser mainly one of speed vs armor.

With the end of World War 1, in defeat the German navy was severely limited, since the UK did not wish to replicate the expensive task of keeping ahead of Germany in another naval arms race (see also the Washington and London Naval treaties more broadly). Initially that meant that Germany could keep only 6 pre-Dreadnought battleships and 6 light cruisers (although, at the time, the definition of light vs heavy to be found in the later naval treaties did not exist). None of those 6 battleships were any match for any existing UK ships. This limit on the Germany navy, like the other armament limits, was considered punitive by Germany.

It was with this backdrop that the Deutschland class were conceived, nominally as a replacement for the out-of-date pre-Dreadnought ships. Since the old battleships were armed with 11-inch guns, the Deutschlands could be as well. Further, by at least pretending to stay under the 10,000 ton cruiser displacement under the London/Washington naval treaties, their design and construction could only cause some discomfort for the UK, but would not obviously abrogate the Versailles treaty.

The punitive nature of the Versailles treaty was at least partially negated by the Anglo-German Naval Agreement in 1935 which, while irritating France, allowed Germany to build up their fleet up to 35% of the existing UK fleet. This agreement was signed on 18 June 1935. Meanwhile Gneisenau had been laid down in May of 1935. She and her sister Scharnhorst were designed, pre Naval Agreement, to also use 11-inch guns even though they would clearly be in violation of the cruiser limits, but perhaps could be argued to be replacements for the old WW-1 battleships.

Certainly, until later (with Bismark and Tirpitz), Germany did not want to risk being seen to either (1) be dramatically violating the Versailles Treaty or the Naval Agreement, or (2) wanting to kick off a new naval arms race. Either might draw more attention to their general re-arming, with consequences. Now, both the Deutschlands and the Scharnhorsts did provoke responses from both the UK and France (for example the Dunkerques).

So - the Scharnhorsts were not classic 'battlecruisers' in the WW-1 sense like Renown was. They were hybrid ships, keeping kind-sorta within various treaty limitations. Further, they had not desire to fight either real battlecruisers or battleships (which the failed at miserably). They could be expected to out duel a single heavy cruiser, and perhaps do a bit better than Graf Spee did against 1 heavy and 2 light cruisers. But, they worked within what they were designed for.

The two ships you mention were built twenty years apart. Yet the German naval architects had no opportunity to practice their craft in those years, so the situation becomes even less comparable.

• Don't just look at the caliber of artillery, look at projectile design, projectile weight and muzzle energy. The German guns fired faster shells but that might not have compensated for the weight difference.
• Armor penetration in "inches" was notoriously dodgy. Against what armor? What range?
• The design decisions of the Renown were influenced by components already at hand, including battleship turrets. She was wartime construction and built in a hurry.
• Armor is equally hard to compare. How thick? How hard? What coverage? But looking just at thickness, the German design had much heavier armor.

Unless, for some reason, a battery of German 11 inch guns was not inferior to a battery of British 15 inch guns?

I think in this case a more holistic look at the armaments of the two vessels will show the answer. Six 15 inch guns on the Renown compared to nine 11 inch guns on the Scharnhorst. The Scharnhorst also had 42 smaller guns, compared to the Renown's 17.

Additionally, the range of the British 15in MkI maxed out around 33 thousand meters, while the 11in SK C/34 had a maximum range of almost 45 thousand meters.

I'm not sure that your supposition that the German ship was inferior in armament to the British is correct.