The ONLY reason we can come boldly into God's presence is because of what Jesus has done.
But why is what he did any different than what the Aaronic priesthood did? Three reasons written plainly in the text:

1. He was tempted like we (as humans) are, but was completely sinless (4:15)
As such, even though he has no weakness, he can still sympathize and "deal with gentleness" with us. (5:2)

2. Because he was without sin, he himself could boldly approach God (he needed no sacrifice for his own sins) (4:15, 5:3)

3. He came from heaven rather than earth. He was fully God, but fully man, too. (4:14)

These are the reasons why, even though the Aaronic priesthood was ordained by God, it was unable to allow even God's chosen people (the Jews) to come before him. Now, through Jesus, anyone who accepts him as their Prophet, Priest and King can not only come, but come boldly before God and receive Grace and Mercy.

Hebrews 4:14-16 and Hebrews 5:1-4