Lightning Components will give much greater reusability of features, separation of concern between components, and are much more opinionated about structure. These are fairly neutral things. Several very good answers have already been given about why to use Lightning Components and Apps.
It is useful to know when you will still use Visualforce.
I need to deliver a custom publisher action, custom button, UI inline to my page layout, or anything in the standard desktop browser UI
There are three ways to surface a Lightning component UI today.
- A tab in the Salesforce1 Mobile app
- A Lightning App Builder app that is surfaced as a tab in the Salesforce1 mobile app (which pretty much gives you the exact same thing today as you get in the first option, only it is in Pilot only)
- A lightning app with its own requestable URL
If your custom UI doesn't fit well into one of these use cases, Visualforce is the way to get it done.
I need to deliver a dynamic email template
Visualforce has a lot of applications outside of just a custom page. One of those is dynamic email templates. This is a powerful feature that is not replicated in Lightning Components.
I need to dynamically generate a PDF
Pretty much the same as above.
The Only Constant is Change
This will all change and evolve over the ensuing months/years as Lightning Components evolves into a much more key part of our UI, but I'd say this is the best characterization of the reality today.
Don't get me wrong, I'm extremely excited about what Lightning Components has to offer, and some of its promise. I just thought it would be a useful to be specific in this answer to complement some of the other very good answers.
----EDIT----
I've updated this answer based on the current state of Lightning Components as of late July 2015.