The difference between 依然 and 仍然 is very subtle. 依然 focuses more on the comparison between now and past; 仍然 emphasizes on "the situation continues to exist":

1. 依然, same as before, same as in the past
E.g. 他虽然已经是总统了，可是对他的小学老师依然很恭敬。 [phr] Although he has been the president, towards his primary school teacher he remains deferential.
This is to say, the respect that he has towards his teacher is the same as before when he was a child.

2. 仍然, indicating there is no change in the situation or the situation continues to exist
E.g. 虽然他已经年过半百，但身体仍然很健康。 [phr] Though more than fifty years old, he is still quite healthy.
This is to say, the situation "he is healthy" does not change and it continues to exist as time passes.