As Randy says, there's no practical difference.
However, there is a slight difference in meaning to me:
I am always running late "I always feel like i am rushing to be somewhere because i am late"
I am always late "it seems that i always arrive late"
Therefore, the first is a thought/complaint about rushing and feeling like you are always late, and the second is a thought about the result of being late. In the second one, perhaps you did not realise you were running late, and therefore arrived late.