If you want to compare the search results from Google and Bing then no need to browse both websites on different tabs. You can have a look at the search results on a single page where both the websites are used as <iframe> and there is no need to open two different windows or tabs.
Since the pages are <iframe>, so you will get the results from Bing and Google server only. <iframe> is just a way to embed a website on different website, the data comes thru the original website only. This way you can compare the keywords related to your blog or website as well.
Screenshot: Google And Bing Search Results
Go ahead and enjoy the features of Google and Bing at the same time. You can watch videos by hovering the mouse on Bing frame as well as you can search the images on Google Image Search.