Notice there are seven things mentioned about each of these mountains, which reveal the characteristics of each:
Hebrews 12:18-21 For ye are not come unto the mount that might be touched (1), and that burned with fire (2), nor unto blackness (3), and darkness (4), and tempest (5), And the sound of a trumpet (6), and the voice of words (7); which voice they that heard intreated that the word should not be spoken to them any more: (For they could not endure that which was commanded, And if so much as a beast touch the mountain, it shall be stoned, or thrust through with a dart: And so terrible was the sight, that Moses said, I exceedingly fear and quake:)
Hebrews 12:22-24 But ye are come unto mount Sion, and unto the city of the living God, the heavenly Jerusalem (1), and to an innumerable company of angels (2), To the general assembly and church of the firstborn (3), which are written in heaven, and to God the Judge of all (4), and to the spirits of just men made perfect (5), And to Jesus the mediator of the new covenant (6), and to the blood of sprinkling (7), that speaketh better things than that of Abel.
The blood of the first covenant (the old covenant, Old Testament) was the blood of goats and calves shed by the mediator Moses - it could only cover (atone for) the sins of the people.
Hebrews 9:18-22 Whereupon neither the first testament was dedicated without blood. For when Moses had spoken every precept to all the people according to the law, he took the blood of calves and of goats, with water, and scarlet wool, and hyssop, and sprinkled both the book, and all the people, Saying, This is the blood of the testament which God hath enjoined unto you. Moreover he sprinkled with blood both the tabernacle, and all the vessels of the ministry. And almost all things are by the law purged with blood; and without shedding of blood is no remission.
The blood of the second covenant (the new covenant, New Testament) was the sinless, precious blood of Christ, the only Mediator between God and man - this not only covered the sins of the people (Romans 5:11), it bore them far away, out of the sight of God forever. This was pictured by the Scapegoat released into the wilderness on the Day of Atonement, and clearly proclaimed by John the Baptist:
John 1:29 The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world.
Hebrews 9:11-12 But Christ being come an high priest of good things to come, by a greater and more perfect tabernacle, not made with hands, that is to say, not of this building; Neither by the blood of goats and calves, but by his own blood he entered in once into the holy place, having obtained eternal redemption for us.
Hebrews 9:23-26 It was therefore necessary that the patterns of things in the heavens should be purified with these; but the heavenly things themselves with better sacrifices than these. For Christ is not entered into the holy places made with hands, which are the figures of the true; but into heaven itself, now to appear in the presence of God for us: Nor yet that he should offer himself often, as the high priest entereth into the holy place every year with blood of others; For then must he often have suffered since the foundation of the world: but now once in the end of the world hath he appeared to put away sin by the sacrifice of himself.
The old covenant brought fear, judgment, condemnation; the new covenant brought forgiveness, peace with God, access into His presence - eternal redemption through the Lord Jesus Christ, of which all the Old Testament types pictured and all the Messianic prophecies pointed to.
I realized that I only touched the tip of the iceberg with this post (just some thoughts based on what I was reading earlier) - of what is surely an inexhaustible subject (from an inexhaustible Book!). If you ever run out of subjects to preach on, try tackling this one for a millenia or two! ;)