The war with the Klingon's timeline is the alternate future. Remember the ENT-C travelled 20 years into the future through a hole in space. Which omitted them from existence. They emerged 20 years in to the future and found out what happened because they never left a mark on history. Travelling back in time corrected things because the ENT-C and her actions became apart of history. Where before their was only an omission.
Think about it in real terms. Example: If the Japanese never attacked the US in 1942 the US wouldn't have entered WWII when they did. We can't know for certain but Germany may have been successful in conquering Europe and the conflcit would have evolved to US vs Germany for the fate of the planet. Omitting the attack on Pearl Harbour would also have affected the development of the A-Bomb by American scientist. Germany was developing their own but Hilter believed more in rockets and missles over a single atomic weapon.
This is a speculative stretch but you can judge how omitting something from history can affect the future reality of something.