The search history card is now called Web & App Activity and deals with searches and browsing activity. In addition to the Google Search history, there's also an option to "include history from Chrome and other apps in your Web & App Activity." If this feature is enabled, Google saves information like your Chrome browsing history, the apps you use and the data they share with Google. "This lets you see things like sites or content that you've looked at in other apps in your Google search results," explains Google.
When you uncheck "include history from Chrome and other apps in your Web & App Activity", Google shows this warning: "Pausing this setting may result in fewer content suggestions in products like Google Now or Google+, and some autocomplete suggestions in Search will not appear. Remember, you can search or browse in incognito mode whenever you want to be more anonymous."
There's a page for Web & App Activity history: https://history.google.com/history/app and it shows the mobile apps you've used on your Android devices. You can remove this information.
Google added 2 new cards for Voice & Audio Activity and Device Information.
"Voice & Audio Activity helps recognize your voice and improve speech recognition by storing your voice and audio inputs to your account (for example, when you say 'Ok Google' to do a voice search)." This feature is automatically turned on by the mobile Google Search app when you enable the always-on OK Google hotword. You can see the audio history here: https://history.google.com/history/audio.
When you pause audio activity, Google shows this message: "Pausing Voice & Audio Activity may limit or disable features such as using 'Ok Google' to start a voice search and reduce the accuracy of speech recognition across Google products that use your voice. Note that this setting does not affect storage of information by Google products (like Voice) that can be used to store your audio or voice inputs. Google may also continue to collect and store audio data in an anonymized way."
There's also a section for device information, which "stores your contacts, calendars, apps, and other device data to improve your experience across Google". You can find the device history here: https://history.google.com/history/device, but it's not very useful. Google shows status messages like: "System state saved", "Contacts saved", "List of installed apps saved".
Click the Pause button and you should see this warning: "Pausing Device Information means you will receive fewer reminders about upcoming events, and features like speech recognition (especially of names in your contact list) may be less reliable. Note that this setting doesn't affect the storage of information by certain other Google products (like Calendar, Contacts, or Play)."
Some of these features were already available, but it's now easier to disable them from Google's account settings.