Update 1
The Neo4J Web UI is built on top of D3.js using SVG: due to SVG performances in a browser when you have more than 500 nodes in a network, the user experience starts to degrade quite quickly.
Handling more than 1000 nodes adds to the technical challenge: in fact with so many nodes what happens most of the time is the "hairball" effect.
This is a blog post that might be useful (disclaimer: I am a developer for KeyLines) about visualizing big network with some design hints.
As you can imagine visualizing more than 1000 nodes is not that easy and that's why some companies such Cambridge Intelligence (KeyLines), Tom Sawyer (Perspective) or Linkourius came up with specific products for that.
You can of course build the visualization yourself for fun with open source libraries but keep in mind that it can take a very long time.
If your Neo4J project is not commercial I can suggest to have a look to Gephi to visualize it: it is a Desktop Application and it has a Neo4J adapter plugin. It can easily handle huge datasets but of course it lacks the same portability of a webapp.
In case you need ONLY a storage for your graph/data than a visualization is not required, you're right.
Original Answer
I think you might have to implement a custom visualization too see such graph in the browser, using one option of those in this page: http://www.neo4j.org/develop/visualize .
Alternatively have a look to this most extensive list here: Big data visualization using "search, show context, and expand on demand" concept
Or maybe have a different visualization approach with one of the following: Data Visualization libraries