It is interesting to note the consistency of scripture in the use of these terms, Rock and stone. Throughout scripture, the Rock is God (Father or Son):
Deut 32:4
4 He is the Rock, His work is perfect; For all His ways are justice, A God of truth and without injustice; Righteous and upright is He.
NKJV
Deut 32:15
Then he forsook God who made him, And scornfully esteemed the Rock of his salvation.
NKJV
Deut 32:18
18 Of the Rock who begot you, you are unmindful, And have forgotten the God who fathered you.
NKJV
Deut 32:30-31
30 How could one chase a thousand, And two put ten thousand to flight, Unless their Rock had sold them, And the LORD had surrendered them? 31 For their rock is not like our Rock,
NKJV
2 Sam 22:47
47 "The LORD lives! Blessed be my Rock! Let God be exalted, The Rock of my salvation!
NKJV
2 Sam 23:3
3 The God of Israel said, The Rock of Israel spoke to me:
NKJV
Ps 18:46
46 The LORD lives! Blessed be my Rock! Let the God of my salvation be exalted.
NKJV
Ps 28:1
To You I will cry, O LORD my Rock:
NKJV
Ps 42:9
9 I will say to God my Rock, "Why have You forgotten me?
NKJV
Ps 95:1
Let us shout joyfully to the Rock of our salvation.
NKJV
Ps 144:1
Blessed be the LORD my Rock,
NKJV
Isa 17:10
10 Because you have forgotten the God of your salvation, And have not been mindful of the Rock of your stronghold,
NKJV
Isa 44:8
Is there a God besides Me? Indeed there is no other Rock; I know not one.'"
NKJV
Hab 1:12
O LORD, You have appointed them for judgment; O Rock, You have marked them for correction.
NKJV
1 Cor 10:4-5
For they drank of that spiritual Rock that followed them, and that Rock was Christ.
NKJV
As for the stone, there is much less, but here is what we do find:
John 1:42
42 And he brought him to Jesus. Now when Jesus looked at him, He said, "You are Simon the son of Jonah. You shall be called Cephas" (which is translated, A Stone).
NKJV
Note this verse above. Some claim that cephas is translated "Rock", but even scripture itself refutes that point quite directly and shows that Cephas means "stone".
1 Peter 2:4-6
4 Coming to Him as to a living stone, rejected indeed by men, but chosen by God and precious, 5 you also, as living stones, are being built up a spiritual house, a holy priesthood, to offer up spiritual sacrifices acceptable to God through Jesus Christ. 6 Therefore it is also contained in the Scripture,
NKJV
So, we are stones, but there are references to Jesus as a stone as well:
Rom 9:33
33 As it is written:
"Behold, I lay in Zion a stumbling stone and rock of offense, And whoever believes on Him will not be put to shame."
NKJV
He is both a Rock and a Stone. That is because he is the cornerstone:
Eph 2:19-22
19 Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God, 20 having been built on the foundation of the apostles and prophets, Jesus Christ Himself being the chief cornerstone, 21 in whom the whole building, being joined together, grows into a holy temple in the Lord, 22 in whom you also are being built together for a dwelling place of God in the Spirit.
NKJV
The cornerstone is in fact, a rock. So Jesus can be called a stone (cornerstone), but is more frequently called the Rock and even the reference to Him being a stone refers to a Rock (cornerstone). On the other hand, there is no reference in scripture anywhere of Peter being called a Rock. He is a stone, as we all are stone per 1 Peter 2:4-6. This may also be a reference to the fact that Jesus is both God (Rock) and man (stone), and is the sole person to hold such a distinction.
Jesus, as the Rock, is also the cornerstone, which is the most notable piece of the foundation, but the confession of Peter that Jesus is Christ is the foundation upon which the church will be built. We see this endorsed in scripture as well, later by Paul:
1 Cor 3:11
11 For no other foundation can anyone lay than that which is laid, which is Jesus Christ.
NKJV
So the foundation is Jesus, not Peter. A church built upon Jesus, and the revelation of the fact that he is the Messiah, the Son of the Living God is the church that will stand, not a church built upon a man.