Just in case that you are using Unity, currently the Unity SDK does not support Cognito User Pool. (That is, AWS hosted list of users) I am currently hesitating because of this. See my post here that they confirmed it is true, currently (26/06/2017) the feature is still not available which might shows a lack of attention to Unity users from them.
However if I use Firebase for logins I would need some more integrations for those credentials to use AWS services. (I would like to use S3 and DynamoDB but only logged in user can use it.) This also made me realize I should move everything to Firebase to save my time and frustrations as soon as possible. (The real time DB is pricier than S3/DynamoDB, but Unity has its own replacement of AWS MobileAnalytics)
AWS S3 recently got nicer UI, which I think is close to Google's level. But other than that I think Firebase's UI is much more joyful to use.
Also, Firebase authentication is free while Cognito is free up to 50k monthly active user. (The next 50k will cost 0.0055 which means if you have 100k MAU it will be 50000 * 0.0055 = 275 USD https://aws.amazon.com/cognito/pricing/)
One more thing, the AWS .NET documentation is a nightmare to read/search in my opinion.
