What is the difference between 几乎 and 差点儿?
差点儿 is more emotional. 

我差点儿迟到了。 implies that I feel lucky I was not late and I would feel bad if I were late.
我几乎迟到了。  states a fact and sounds like it's not a big deal even if I was late.

Visualize such a scenario. I run across the road and a car nearly hits me. My girlfriend says 你差点儿就被撞了！ It's a blame but I can feel she's really worried and feels grateful I was not hit by the car. I would be really sad if she says 你几乎就被撞了。