Okay, can’t pass up this matchup.
Speed, of course, is the same, so it doesn’t play a factor but we’ll assume they’re charging at each other in empty ocean on a clear day.
So there are two factors: Whether Scharnhorst has torpedos and 15 inch guns.
To give the ship the best chance, let’s say Scharnhorst does have them.
Hood’s broadside is still bigger than Scharnhorst’s but German 15 inch guns have longer range.
Scharnhorst has more torpedos, not a game changer but it’ll help.
Scharnhorst is also more well armoured compared to Hood.
Finally, a rather flimsy statement can be made with the Battle of Denmark Strait.
It appears that the British cruisers were eventually involved in the battle. If not, it gets closer.
So the British lose two cruisers and a battleship.
Germany only loses 1 cruiser and half a battleship (Scharnhorst is weaker than Bismarck).
So the British lost Hood even with a supposedly superior material advantage, and more is taken away from the British than the Germans.
So maybe, just maybe, the odds are better with Germany than in Denmark Strait (this does not count tactical considerations like position and the malfunctioning of the guns of Prince of Wales).
And with the same German guns, there’s always the chance of the lucky hit on the Hood repeating itself.
I’ll give this to Scharnhorst. If you give it 11 inch guns and no torpedos, it’ll be closer.