The search network is made up of users actively looking (i.e. searching) for goods, services, and information. This express intent by the user means that your search ads are focused around informing the user about your options so the user can take action.
With the search network, you can user Expanded Text Ads, Dynamic Search Ads, or Responsive Search Ads.
The display network is made of up publisher sites that show ads. The user's primary purpose on that page is to watch a video or read an article. Your ad needs to draw attention to itself so the user wants to interact with your ad and visit your site.
The display network has specialty ad formats that make the ads more visible, such as image ads or Responsive Display Ads.
As the user intent and the networks are so different from each other, it's best to run your ads only on search or only on the display network and not combine the two.
When you have a campaign showing on both networks; you can see an alert on your dashboard.
When you click on the alert; you'll see the campaign screen filtered by the alert. Click on the gear icon to open that's campaign's settings:
On the settings tab; open the network portion of your settings and you can click on a button to only run this campaign on the search network, which will disable that campaign from showing on the display networks.