Difference between Leaking Factor and Forgetting Factor
The play similar role in those algorithms - the ability to forget the past and adapt to current reality.  

In the LMS, the classic implementation has $ \alpha = 1 $.
Namely the optimal weights at any point are function of all inputs.  

The Leakage factor allows to weigh the past differently in a damped manner which over times means the far past has practically no significance on the current result.
The other side means the ability to integrate data against noise is decreased which is the same balance the user stands when selecting the parameters of the RLS.

Namely, as usual, the balance between high bandwidth filter which adapts fast yet is sensitive to noise or the low bandwidth filter which is slow yet if the data is stationary can handle high energy noise.