Android: API level VS. Android version
Well, API is for development, so the changes in new API version are more "inside". But new version of Android usually adds more features for users, that are "visible".
Check this page http://developer.android.com/guide/appendix/api-levels.html, there is a table that shows relations between versions and API levels.
Multiple versions of Android can have the same API level but the API as an integer allows developers to more easily target devices. The chart below will give you an idea of their relationship but only the documentation contains exhaustive listings of the API levels and how they differ from each other.
Source: developer.android.com.
Because this data is gathered from the new Google Play Store app, which supports Android 2.2 and above, devices running older versions are not included. However, in August, 2013, versions older than Android 2.2 accounted for about 1% of devices that checked in to Google servers (not those that actually visited Google Play Store).
In addition to the answers provided, there is a detailed explanation of the Android Platform usage on Wikipedia (permalink).
This table will give you a highlight of Android API vs Version.
