On the ZODB:
Another way to ask "What is the real purpose of the ZODB?" is to ask, "Why was the ZODB originally created?"
The answer to that is the project was started very early on, around 1996. This was before the existance of MySQL or PostgreSQL, when miniSQL (a free-to-use but not free software) database was still in common use, or big money databases such as Oracle. Python provided the pickle module to serialize Python objects to disk - but serialization is lower level, it doesn't allow for features such as transactions, concurrent writes, and replication. This is what the ZODB provides.
It's still in use today in Zope because it works well. If you have no existing skillset in realational databases, it's easier to learn to use the ZODB than a relational database. It's also usable simpler use-cases, for example if you have a command-line script that needs to store some configuration information, using a relational database means having to run a database server just to store a little bit of configuraiton. You could use a config file, but the ZODB also works quite nicely because it's an embedable database. That means that the database is running in the same process as the rest of your Python code.
It's also worth noting that the API used to store objects inside containers is different between Zope 2 and Zope 3. In Zope 2, containers are stored as attributes:
root.mycontainer.myattr
In Zope 3, they use the same interface as Python standard dictionary type:
root['mycontainer']myattr
This is another reason why it can be easier to learn to use the ZODB than the Django ORM, since Django has it's own interface for it's ORM which is distinct from Python's existing interfaces.
Through-the-web (TTW):
Again, understanding the reason for TTW goes back when Zope was developed. While it seems silly to break with well known developer tools such Subversion or Mercurial, Zope was developed in the late 90s when the only free version control system was CVS. Zope 2 had it's own simple version control capabilites, and they were as good as CVS (which is to say, "they were limited and sucky."). UNIX workstations cost a lot more money back then, and had far fewer resources, so System Administrators were much more guarded and careful about how servers were managed. TTW allowed people who might not normally be able to upload code to the server with sysadmin intervation a way to do that.
With text editors, emacs and vi have had ftp-modes, and Zope 2 can listen on an FTP port. This would allow you to develop so that code was stored in the ZODB (editable TTW), but it was common to edit this code using a emacs or vi.
Today in Zope, TTW is more rarely used or promoted since it no longer makes sense to do this. Disk space is cheap, servers are (relatively) cheap, and there are lots of developer tools which expect to interact with the standard filesystem.
Acquisition:
It was a mistake. It was a very confusing feature that caused lots of unexpected things to happen. In theory there are some interesting ideas to acquisition, but in practice it's best tossed in the bin and has little practical use.
Moving from Django to Zope:
Work started on Zope 3 in 2001. This fixed a lot of the problems with Zope 2. It's a testament to the Zope community that Zope 2 is still actively and well maintained, but it's hardly state-of-the-art. Zope 2 is really only interesting to learn from a historical perspective.
Zope 3 ended up getting evolved in a few different directions, and so modern incarnations of Zope are best expressed in the form of Grok, BFG or Bobo.
Grok is closest to Zope 3, and as such is a pretty large framework - it can be rather overwhelming at times when delving through it's code base. However, just like Django, or any other full-stack framework you don't need to use every part of Grok, it can be quite easy to learn the basic and create web applications with it. It's convention-over-configuration is second to none, and it's class-based Views give it a much tighter, arguably cleaner code base than a Django web application. It's URL routing system is extremely flexible, but also arguably over-engineered.
BFG is a "pay for only what you eat" framework written by long time Zope developer Chris McDonough. As such, it's closer to Pylons in spirit, where only the parts deemed core or essential to a framework are included. It also plays very well with WSGI. It only uses a few core Zope packages.
Bobo is a "micro-framework". It's just a way to route URLs and serve up an app. It doesn't use any Zope packages, so isn't strictly in the Zope family of web frameworks. But it was written by Zope's creator, Jim Fulton, who originally called the publishing part of Zope, "Bobo". The original Bobo, written in the early 90's, mapped URLs to packages and modules, so if your source code was layed out as:
mypackage.mymodule.MyClass
You could have a URL such as:
/mypackage/mymodule/MyClass
Which was very inflexible, and was replaced with URL Traversel in Zope 2, which is fairly complex. Bobo uses Routes, so it's a middle ground between dead-simple URL resolution and complex URL resolution - about the same in complexity as Django's URL resolution machinery.