In general, I think in your case, you are likely to see some benefits from persistent connections. There are also drawbacks, but these are manageable as long as you keep them in mind. You may, however, want to go further and consider an actual connection pooler.
The big issue is that typically PostgreSQL does best when concurrent connections are under approx 2 per CPU core plus one per disk spindle (due to I/O wait time). This isn't exact of course but it gives you an idea given your hardware and resources what to expect.
The connection startup/teardown overhead is not that much on Linux/UNIX platforms, but managing concurrency may be critical to keeping things running fast. So I would start out with persistent connections and then move to a connection pooler if I needed some additional control there.
There major disadvantages are that there are certain database operations which you cannot do when others are connected to the database. If you ever need to restore from backup, you may need to make sure you disconnect the web app first.