You can replace one video for another inside of Vidalytics and have the new one show up on your site without having to touch your embed code (yay!). 

To do this, go to your video settings page and click on the Settings icon in the top right corner.
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Next, click the Replace Vid dropdown.

Select which video you want to replace the current one with and click on Replace Selected Vid With This One.
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The replacement process will begin in the background.

Make sure you hit the Publish button on your page, and allow up to 15 minutes for your new video to show up on any pages you had the old one embedded. 

Your old video will remain in your account, so you can keep its data. 

Note: replacing a video won't keep the current settings of the original. Make sure your replacement video has all the settings you want before following this process, or go to your replacement video's page and tweak the settings over there and republish after the replacement has taken place.