How can you compare both...Route 53 is a high available DNS service with versatility and lot of options where CloudFlare is more like a web proxy/filter. So they are serving from different levels except the fact that both can help to scale-up/down your infrastructure and can help to setup failover or so!
To get an accurate comparison you should be comparing Cloudflare to Amazon's Route 53, Cloudfront (with WAF rules).
Depending on where you are in the world and how much you want to spend will likely be your deciding factors.
Assuming your not using any AWS services and this is just for a "basic" website and your website is targeted to users in North America, Cloudflare's $20 business plan will provide website security and DNS hosting for your site with no extra usage limits. However Cloudflare doesn't peer with all ISPs in all locations so Cloudflare is not great for Australia / NZ / Oceania.
For the same $20 you'll get about 100GB of traffic on AWS's Route 53 & Cloudfront and the performance should be good globally and in Australia / NZ / Oceania...
AWS products are more complicated to setup, Cloudflare is click a grey cloud to enable Cloudflare features.
Amazon Route 53 is just an authoritative DNS service while Cloudflare is a website accelerator optimizing the delivery of web pages to your end users and improving the security of your website. You can think of the functionality offered by Amazon Route 53 only a cog in the wheel.
Cloudflare is pretty much Amazon Route53 (with Anycast) + Dynamic content being served through Amazon EC2 instances from multiple regions (Cloudflare uses a proxy approach; so your content is actually still severed from the same hosting provider) + Amazon CloudFront for static content + Custom threat/security scan layer
I would say that Amazon has become too greedy and they want to step foot in each and every inch. Having said that, I would suggest anyone to go Cloudflare. Amazon want to create an ecosystem, to bring you in and the start charging you for everything.