The first, ἀμνός is used by John the Baptist to identify the Lamb of God:
The next day he saw Jesus coming toward him, and said, “Behold, the Lamb of God, who takes away the sin of the world! (John 1:29 ESV)
and he looked at Jesus as he walked by and said, “Behold, the Lamb of God!” (John 1:36)
John identified Jesus, using the word ἀμνός as did earlier NT writers, Luke and Peter. Many commentators understand Acts 8:32 and 1 Peter 1:19 are referring to Isaiah 53:7, in which the LXX also uses ἀμνός.
The second, ἀρνίον is by Jesus to identify His lambs:
15 When they had finished breakfast, Jesus said to Simon Peter, “Simon, son of John, do you love me more than these?” He said to him, “Yes, Lord; you know that I love you.” He said to him, “Feed my lambs.” 16 He said to him a second time, “Simon, son of John, do you love me?” He said to him, “Yes, Lord; you know that I love you.” He said to him, “Tend my sheep.” 17 He said to him the third time, “Simon, son of John, do you love me?” Peter was grieved because he said to him the third time, “Do you love me?” and he said to him, “Lord, you know everything; you know that I love you.” Jesus said to him, “Feed my sheep. (John 21)
When Jesus reinstates Peter, His first instruction is to feed His lambs, ἀρνίον. The second is to tend to His sheep, πρόβατά and the third is to feed His sheep, also πρόβατά. Jesus' use of ἀρνίον for lamb makes sense. First, it is different from ἀμνός and so distinguishes Himself (the Lamb of God) from His lambs. Second, it makes for a logical progression. Peter's work is threefold. Start by feeding the lambs. A ἀρνίον, is literally a little lamb and would be the least able to care for it itself. Then tend, or shepherd the sheep, and then feed the sheep.
In Revelation Jesus is revealed as The Lamb Who Was Slain and the word used is ἀρνίον:
saying with a loud voice, “Worthy is the Lamb (ἀρνίον) who was slain, to receive power and wealth and wisdom and might and honor and glory and blessing!” (Revelation 5:12)
In fact, whenever "Lamb" is used in Revelation it is always, ἀρνίον and ἀμνός is never used.
Why call the Lamb of Revelation ἀρνίον, which is the lamb Peter is to feed and not ἀμνός, which is the Lamb who takes away the sin of the world?
Note: since ἀμνός is the diminutive of ἀρήν, my question originally asked about the absence of both ἀρήν and ἀμνός. It has been modified to focus strictly on the difference between the terms as they are used in the Fourth Gospel.