how to compare the designs of chanel and vivienne westwood?

I think the first thing you should mention is the difference in Fashion Eras of both the designers because time dictates fashion styles…
Gabriella ‘Coco’ Chanel entered the industry around 1912 when women were suffering the constraints of the corset…She offered the solution of relaxed dressing which was corset free with blazers.skirts,suits and ‘the little black dress’ and introduced jersey wear. She was also a milliner and began her business in this way…
Design secrets such as weighing hems of skirts with internal chains sewn into the hems were created by her, and are still used today.
Chanel’s clothes were timeless classics, along with her classic perfumes.

Vivienne Westwood entered the fashion scene in the Seventies and introduced the punk look to British audiences, inspired by the punk music which was being popularised in America at the time eg The Ramones. .
Her then boyfriend Malcolm Mclaren introduced The Sex Pistols into the Music industry,who wore
her statement pieces, created to shock in a very contrived way…They were not intended to be timeless classics, more throwaway with usage of bin bags, T-Shirts, and safety pins…

Chanel was French whilst Westwood is English…
Chanel is no longer alive whilst Westwood is very much alive and kicking…

Westwood has, over the years, changed her stance on Fashion by producing classic corseted pieces,inspired by British History, quite the opposite of her original ‘God Save The Queen’ T-Shirts complete with a safety pin piercing the nose of the Queen’s face depicted on them…Corseting features heavily in her current collections…

They were both helped by the men in their lives…

Westwood has also introduced her own perfume alongside many other products such as jewellery, her ‘rocking horse’ shoes etc

You might enjoy the film version of Chanel’s life story ‘Coco, Before Chanel…

Good Luck!