Company name in a contact record is a default single-line text property that a HubSpot user can update manually. This property is separate from any company associations that the contact records has and is not affected by automatic association or manual association.
Associated company as a column in the contacts dashboard shows the Name property of the associated company that the contact is associated to, and is directly affected by automatic association or manual association with the company record. In the contact record, you can see the associated company in the right panel under the Company section.