What is difference between lightning app and connected app in Salesforce?

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 .
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 .

Lightning App: The Lightning App Builder is a point-and-click tool that makes it easy to create custom pages for Salesforce and Lightning Experience. With the Lightning App Builder, you can combine various components on a single page to give your users what they need all in one place.
Connected App: It is an application that can connect to Salesforce over Identity and Data APIs. Connected Apps use the standard OAuth 2.0 protocol to authenticate, provide Single Sign-On, and acquire access tokens for use with Salesforce APIs.