Apple thought - or at least claimed - that the Woolworth's logo was too similar to its own. Since Woolworths application included use of the trademark in Class 9 (which covers data processing equipment and computers, mobile phone covers and accessories, computer devices and computer peripheral devices, telephones, and computer hardware and software) it's not hard to see why Apple objected.
But Woolworths was granted trade mark registration number 1258297, so it seems Apple lost the argument.
Now the Apple is taking on a much smaller business - there's a privately owned cafe in Bonn, Germany called Apfelkind ('Apple child'), which has applied to register a trademark consisting of a picture of an apple with a child's silhouetted face:
Apple has reportedly taken exception, suggesting the similarity between the two logos could confuse consumers. Well, yes - trademarks have to be defended, and 'confusion' (actual or likely) is the basis for allegations of trademark infringement.
Apple's lawyers used Mr Justice Foster's "a moron in a hurry" test when arguing that the then Apple Computer logo couldn't be confused with that of Apple Corps, but they don't seem to apply it when they come across other companies' trademarks that include an apple element.