First of all, the Sodenbergh version (2002) is over an hour shorter than the Tarkovsky version (1972), but it has a much faster pace. Both versions share many of the same scenes, but there are some differences.
Just as the 2002 version, the 1972 version starts with Kris (not Chris) Kelvin's life on Earth. However, instead of showing his day-to-day routine and him being summoned to Solaris, Kris is already preparing to go to Solaris, and spends his last day on Earth saying goodbye to his Earth life and his father at his childhood home. There he is visited by Berton, a scientist who was once stationed at the space station orbiting Solaris. Berton warns him about some strange effects of the planet, by relating a story of one of his survey missions of the planet; Berton saw how the planet manifested the image of a child, who was the son of a colleague who was lost on Solaris during an earlier mission.
While arriving at the space station, Kris' first encounter is with an unknown woman, whom he follows into the morgue where he finds his friend (also called Dr. Gibarian) dead. There is no blood, but the station is heavily neglected. The two remaining scientists are Dr. Snaut (Snow in the 2002 version) and Dr. Sartorius (male in the 1972 version, female Dr. Gordon in the 2002 version). Dr. Sartorius is as reclusive and suspicious as Dr. Gordon, but he and Kris are not as openly antagonistic towards each other. Sartorius' 'Visitor', a small dwarf, is briefly seen.
The 2002 version mentions an earlier intervention team that travelled to Solaris, and wasn't heard from again. There are two bodies in the morgue. According to Snow, the second belonged to a third scientist who was killed by the intervention team when he refused to leave the station, explaining all the blood.
Kris' first encounter with his wife (called Chari instead of Rheya) is practically the same: Kris panicks, lures her into a rocket (not a shuttle) and sends her flying into space. Snaut then tells him about the 'Visitors' each scientist has encountered onboard, Gibarian's apparently being the unknown woman. Gibarian's son does not feature in the 1972 version at all.
In the 1972 version, after Chari's second appearance, Kris accepts her presence, but then locks her inside his room. She panicks and tears through the metal, severely injuring herself, but the injuries heal before his eyes. The door incident occurs much later in the 2002 version, and is combined with Rheya's failed suicide attempt by liquid oxygen.
The Rheya in the 2002 version gets much more exposition: she suffered from depressions and a tough childhood, and terminated a pregnancy without informing Chris; a furious Chris then left her, causing her to commit suicide. Most of this background information is not in the 1972 version.
After Chari's accident, all aboard have several meetings and a birthday party where they discuss the degree of humanity of the Visitors, how they are created and how to get rid of them (largely condensed into one meeting in the 2002 version, where the scientific theory and weapon to permanently dissolve the Visitors differ slightly); however, no one reveals to Chari that Kris killed the first version of her.
Several more technical talks and philosophical conversations about the nature of humanity, love and suffering occur in the 1972 version. Snaut sends Kelvin's brain patterns towards the planet, in the hope that it will understand and stop creating the disturbing manifestations; Sartorius, however, proposes to attack the planet with radiation.
There are no elaborate dreams and flashbacks of Kris' time on Earth before Chari's suicide; instead, Kris shows her some home videos of himself as a child, and of her during their marriage, during which she starts to realize she only has Chari's memories, but not the emotions that normally accompany them. She later learns from Sartorius that the real Chari committed suicide, forcing Kris to tell her the story. After realizing she is not human but merely a copy, she attempts suicide by liquid oxygen, but is regenerated. These sequences are somewhat different in the 2002 version: Rheya remembers her suicide by herself. Gordon is much more paranoid and distrusting of the Visitors; she reveals to Rheya that Chris killed the first Visitor, and berates him for getting emotionally attached to the second. Rheya escapes later that evening, attempts suicide and is regenerated. It is revealed that both Gordon and Snow have witnessed the death and resurrection of other Visitors as well. Chris locks himself in his room and tries to stay awake in order to protect Rheya from Gordon.
Kris has no dream about Gibarian telling him that they may be misinterpreting Solaris' intentions in the 1972 version. Instead, he has a dream about his mother who is concerned about his emotional state. He falls asleep and after waking up, he learns that Chari has escaped again and requested the others to permanently kill her with the weapon; she has left him a suicide note (a similar course of events happens in the 2002 version). Snaut mentions that since sending Kris' brain patterns, the Visitors have stopped appearing; instead, there are islands forming on the surface of Solaris.
One of the biggest differences is the revelation in the 2002 version that the real Snow has been dead for a while; a Visitor killed the real Snow in self-defense and now pretends to be him. This Visitor warns Gordon and Chris that the station is being pulled into Solaris, and that they should leave. Chris, however, chooses to remain at the station with Gibarian's son, as the planet finally dissolves the station. In the ambiguous epilogue, we find him back in his Earth environment, but this time, he has the capacity to regenerate, just like a Visitor. He is joined by Rheya, and they appear happy now. In the 1972 version, we find Kris back on Earth at his father's house, but the final shot shows the house to be situated on one of the islands on Solaris, revealing that Kris has chosen to stay on a manifestation of Earth.