While the phrase "bench warrant" and "warrant" are similar terms, they have two separate meanings. In this context bench refers to where a judge sits, not a bench in the park.
Definition of a Bench Warrant
According to Nolo.com, you should think of this mnemonic device when trying to remember the difference between the two warrants; a bench warrant means someone isn't sitting on a bench in front of the judge where she belongs. If a person has failed to come to court or done anything resulting in contempt of court, the judge can issue a bench warrant for his immediate arrest.
Definition of a Warrant
An arrest warrant is issued by a police officer, not a judge. The officer must have probable cause to believe that the person in question has committed a crime. Arrest warrants are not required for most crimes, but they are required if the police officer wants to pick up the suspect at her house.
The primary differences between the two kinds of warrants is that a judge starts the process in executing the bench warrant because the person failed to show up to court. In an arrest warrant, the police officer starts the warrant execution process by requesting that a judge allows him to arrest a suspect.