Pro Chrome notifications act as a notification drawer
Desktop Notifications uses Chrome notifications to act as a running list of past notification that have not yet been dismissed. This is especially useful if you receive a lot of notifications or you are not at your computer when they are received.
Con Slight delay
While other services like Pushbullet receive notifications almost immediately. There is a delay of a few seconds with Desktop Notifications. Although the developer is working on it. In response to a user review the developer stated "We had to introduce a delay to the notifications because of the overwhelming amount of users who started using our app recently. Otherwise our servers would melt. :) When you don't receive a notification for some time our app will check less frequently. We don't like this either and are working on it!"