Advice needed on Telephoto purchase

I am currently using an APS-C Canon DSLR. I am looking to buy a zoom/telephoto lens since I don't currently own a lense in the longer focal length. I am happy to buy used and have a max budget of around £600/$825.

I have narrowed it down to the Canon EF 70-200mm f/4 L IS USM and the Canon EF 70-300mm f/4-5.6 IS ii USM lens.

The condition of the lenses being equal if I buy used, the 70-200mm is about £120/$150 more expensive. I intend to use it for punching into landscapes, wildlife and some sports.

My main question is... Is the up in image quality and build quality of an L series lens worth the extra money? Or have I got it all wrong and there is a different lens I should be looking at?

Thanks for the help guys

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