As Apple has already published the update on the Mac App Store for users who have the first Public Beta build of Mac OS X 10.11 El Capitan installed on their Macs.
Unfortunately, there are no release notes for the second Public Beta build of Mac OS X El Capitan 10.11, except for a very brief line of text that recommends it to all users. Most probably it patches some of the most annoying issues users have encountered since the first Public Beta build, and we can tell you that there were a few.

How to install OS X 10.11 "El Capitan" Public Beta

To install the first Public Beta build of OS X 10.11 "El Capitan," you will have to go to the Apple Beta Software Program website, sign in with the same Apple ID that you use on your Mac, and read all the information posted on the OS X "El Capitan" Public Beta page, especially the part where you will have to back up all you important data.
You will have to enroll your Mac to get the Public Beta of OS X 10.11, so click the "enroll your Mac" link on the "Getting Started" section. You'll be automatically redirected to another page, where you must follow the steps to install "El Capitan." Then, you need to click on the blue "Download the OS X El Capitan Public Beta" button.
The Mac App Store app will open, and your unique redemption code for installing the OS X 10.11 (El Capitan) Public Beta build will be activated automatically and the installer will begin to download in the background. Please note that it has 6GB, and it might take a while, depending on your Internet connection.
You can see the download progress in the Launchpad. When it finishes, the OS X 10.11 Public Beta installer will open, and all you have to do is follow the instructions on the installation wizard to install "El Capitan." After installation, you will be able to update your OS X 10.11 installation when new Public Beta builds are released directly from the Mac App Store.
For those of you who already have the first Public Beta build of OS X 10.11 El Capitan installed, we will summarize again the entire update procedure. First of all, you should open the Mac App Store app, go to the Updates tab, and wait for the "OS X El Capitan Public Beta 2" update to appear.
Click the "Update" button in front of it, then click on the "Download & Restart" button on the pop-up dialog that will appear, and wait for the update to be downloaded. After a successful download process, which might take a while, depending on your Internet connection, your Mac will restart and the OS X 10.11 El Capitan Public Beta 2 build will be installed.