The Beatles' record label Apple Corps has accused Apple Computer in court of breaching a trademark agreement by selling music.
The Beatles set up Apple Corps in 1968
Both companies reached a deal over the use of the apple trademark in 1991, which stopped Apple Computer from entering the music business.
But Apple Corps claims the US company's online iTunes Music Store has broken the agreement.
The hearing opened at the High Court in London on Wednesday.
Apple Corps, owned by former Beatles stars Sir Paul McCartney, Ringo Starr and the widows of John Lennon and George Harrison, was founded in 1968.
Apple Computer, the firm whose home computers helped launch the personal computer industry, was founded in 1976.
Geoffrey Vos QC, representing Apple Corps, said the computer firm had been keen to use the Apple brand on its Music Store and offered to buy the rights from Apple Corps for $1m (£576,500) just before iTunes launched.
The offer was rejected by Neil Aspinall, Apple Corps' managing director.
Apple Computer was founded by Steve Jobs and Steve Wozniak
Mr Vos claimed Apple Computer founder Steve Jobs had said downloading music from the internet was now no different from buying a record.
He argued the US computing giant violated the deal by selling music online and its argument that it used the apple mark only in connection with a delivery system was "plainly wrong".
Mr Vos told the court that calling iTunes a mere electronic device was a "perversion" of the 1991 deal.
He added that the Apple logo was prominent on the iTunes website and almost every advert for it carried the logo.
Mr Vos said: "[Apple] Computer was promoting a store at which to buy music, and more particularly, Computer's musical recordings - permanent downloads - with special characteristics.
"No objective onlooker could think otherwise."
Sir Paul McCartney is credited with dreaming up the Apple Corps name
Mr Vos demonstrated to the judge, a self-confessed iPod owner, how to download from music store.
He chose the 1978 disco hit Le Freak by Chic which sounded in the courtroom as he pointed out to the judge how many times the "apple" logo appeared on the website as he went through the procedure on the equipment set up in the courtroom.
The computer company's logo is an apple with a section removed out of the side. The record company is represented by a complete green Granny Smith apple.
An agreement between the two companies to share use of the Apple trademark was first established in 1981.
But as Apple Computer's business increasingly entered the world of entertainment, the company sought a less restrictive trademark agreement and a court battle ensued in 1989.
Details of the eventual deal, thrashed out in London's High Court over two years, were never disclosed - but Apple Corps was believed to have emerged with about $30m (£17m) from the computer firm.
The launch of the iPod, a portable music player, in 2001, and the launch of its iTunes music store two years later has prompted the latest battle between the two firms.
About three million songs are downloaded every day from the service.
Tracks by The Beatles have not been licensed for downloading and are not available on the service.
Apple Corps is seeking to reinstate the 1991 deal and receive financial damages.