Difference between lightning app and connected app?

Connected application is basically a mechanism to iframe your external application in a secure manner inside Salesforce, allowing you to securely post and get data from Salesforce into your application .

Essentially you do not build UI layer into Salesforce and instead your web app UI renders within Salesforce ui as an iframe .

Lightning components is a JavaScript framework that you use to build components which can be surfaced to users using the application builder.

For lightning components your application ui resides inside the platform.You have JavaScript controller associated with your component and that allows you to hook to Salesforce server side logic.For your use case of you we're to connect to your web app from lightning components ,you will need to expose a serve API that Salesforce server can call to .

You can make your canvas application look like SFDC by following SLDS design patterns and then use force:canvasApp tag to surface your application as lightning component so that it becomes available to the lightning application builder .