When you book through a third party, you lose any flexibility and also cannot deal directly with the airline for changes, cancellations, refunds, etc.
As @sam_gordon said above, always best to book direct.
I always book direct. Those travel websites are simply a form of 'search engine' displaying costs they grab/consolidate from the airline websites. Their main benefit is allowing for easy comparison of available costs/flight times/non-stop options/etc. for your intended travel. None of them to my knowledge off a price that is any lower compared to booking directly using the airline website and more can go wrong working through a 3rd party.
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