How to get Step Response from an Impulse Response in Simulink?
Your system H(s) is continuous

$$ H(s) = \frac{s}{s+1} $$

Using a discrete impulse does not make a lot of sense. I suspect Simulink added a "zero-order hold" or ZOH block between your discrete impulse and your system $H(s)$. As such, your impulse response will be affected by the sampling period of your discrete impulse source. If you decrease the sampling period, the second curve shoud match the step response more closely, although they will never be exactly the same.

Edit : 

Decrease the width of your impulse to say 0.1 second and increase the amplitude to 10 so that the area under the curve stays 1.