If we are talking about interrogation, rather than police interviews with witnesses, then there is one main purpose of having a written statement. It is to commit the suspect to the statements that he or she makes while being interviewed.
There are also some other benefits to a written statement. It can serve as a record of what was talked about. This allows the police and prosecutors to easily go back and closely examine what the suspect has said. However, this is not the main purpose.
The main purpose of a written statement is to get a suspect to formally assert the truth of the things that he or she says. This can be useful for at least two reasons. First, if the suspect makes admissions that tend to show they are guilty, the written statement puts those admissions on record and can be used to help convict the suspect. Secondly, even if the suspect does not make such admissions, the written statement can be used at a trial. If the defendant takes the stand in the trial, the written statement can be used to catch them in instances where they have changed their stories. This will help prosecutors impeach the defendant’s testimony.