Really the only difference is the default ability associated with each skill. The Acrobatics examples in the Player's Handbook could easily be thought of as Athletics (Dexterity). Splitting it into two skills serves to disambiguate which ability score is more likely to be required for a task, in a similar way as the split between Investigation (Intelligence) and Perception (Wisdom). It also has the effect of requiring separate proficiencies, since the training required for Strength- and Dexterity-oriented athletics is somewhat different (despite having some overlap in the real world).

Really the only difference is the default ability associated with each skill. The Acrobatics examples in the Player's Handbook could easily be thought of as Athletics (Dexterity). Splitting it into two skills serves to disambiguate which ability score is more likely to be required for a task, in a similar way as the split between Investigation (Intelligence) and Perception (Wisdom). It also has the effect of requiring separate proficiencies, since the training required for Strength- and Dexterity-oriented athletics is somewhat different (despite having overlap in the real world).