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.