What is the difference between unmanaged vs managed in-app product? What is the difference between a "managed in-app product" and an in-app product that is not managed?
The short answer is that managed items are items that can only be purchased once per user and that Google Play keeps track of their purchase, unmanaged items aren't tracked by Google Play and they can be purchased multiple times.
A fuller description is given in Google's In-app Billing Version 2:

Managed per user account — Items that can be purchased only once per
user account on Google Play. When a user purchases an item that uses
the &quot;managed per user account&quot; product type, Google Play permanently
stores the transaction information for each item on a per-user basis.
This enables you to later query Google Play to restore the state of
the items a specific user has purchased. If a user attempts to
purchase a managed item that has already been purchased, Google Play
prevents the user from purchasing the item again and displays an &quot;Item
already purchased&quot; error.
The &quot;managed&quot; product type is useful if you are selling items such as
game levels or application features. These items are not transient and
usually need to be restored whenever a user reinstalls your
application, wipes the data on their device, or installs your
application on a new device.
Unmanaged — Items that do not have their transaction information
stored on Google Play. This means that you cannot later query Google
Play to retrieve transaction information for those items. For
&quot;unmanaged&quot; purchases, you are responsible for managing the
transaction information. Also, Google Play does not attempt to prevent
the user from purchasing an item multiple times if it uses the
&quot;unmanaged&quot; product type. It's up to you to control how many times an
unmanaged item can be purchased.
The &quot;unmanaged&quot; product type is
useful if you are selling consumable items, such as fuel or magic
spells. These items are consumed within your application and are
usually purchased multiple times.

Note that only version 2 of the in-app billing deals with both managed and unmanaged, version 3 only deals with the managed in-app product type.