Have had many years of suffering
Are singled out by Jesus for healing
Are healed in association with a pool in Jerusalem
Are healed on the Sabbath, resulting in controversy
Talk to the Pharisees about the healing
Are sought out by Jesus after the healing
Healings result in a discourse by Jesus about his identity (John 5:17-47, 9:35-10:30)
But there are some significant differences as well:
The lame man hopes to be healed at a well-known healing pool (Bethzatha), but Jesus heals him by declaration; the blind man is sent for healing to a pool not known for healing (Siloam). Siloam was known as the source of the water used in the water ceremony at the Feast of Tabernacles. This is the same ceremony at which Jesus had declared himself to be the source of living water (John 7:37-39). In one healing, Jesus rejects a superstitious (or at least manipulative) method of healing; in the second, Jesus reminds us that he is the source of living water.
When the lame man talks to the Pharisees, John subtly portrays it as a betrayal by a spiritually dim man. When the blind man talks to the Pharisees, John portrays it as an act of belief, courage and even cleverness. Jesus has no reason to be at the trial, because the blind man capably uses all the sorts of arguments that Jesus would use to prove that Jesus must be from God.
When Jesus seeks out the lame man, he warns him about his sin (John 5:14). But when Jesus seeks out the blind man, he reveals his identity as the Son of Man, (John 9:35-39); he defends the blind man as one who truly sees (John 9:39-41); and even hints that the blind man is one of his true sheep, who hears the voice of the Shepherd (John 10:4-5, 27).
Jesus' warning to the lame man, "Don't sin any more, so that nothing worse may happen to you," affirms his agreement with the Jewish belief that some suffering results from sin. But when Jesus sees the blind man, he reveals something new: "Neither this man nor his father sinned; he was born blind so that the works of God might be revealed in him" (John 9:3). Jesus reminds us that suffering is not necessarily because of something but for something.
Both healing scenes are an opportunity for John to reveal something about Jesus. In John 5, Jesus teaches that he can give life to whomever he wishes (as he picked out the lame man) - both now and on the last day. In John 9-10, Jesus is revealed as the source of light and the good shepherd, in contrast to the leaders of Jerusalem.
The sharpest contrast between the two scenes is about discipleship. The lame man is uncertain if he wants healing, has no recognition of who heals him, and informs the Pharisees as soon as he knows who broke the Sabbath by healing him. There is no evidence that he believes or comes closer to belief. In contrast, the blind man begins the scene as an innocent man. At Jesus' command, he leaves his begging post and walks across town to Siloam (surely an act of faith for a blind man). Before he ever sees Jesus, he testifies in defense of Jesus, refuses to back down, is labelled a disciple of Jesus by Jesus' enemies, and suffers rejection. When he learns more about Jesus, he believes even more, and worships Jesus (John 9:38).
The picture: Healing of the Blind Man, by Duccio di Buoninsegna, ca. 1308.