When you embed a YouTube or any other 3rd party video, your visitor's browser will need to make a number of extra requests to render your page.
With YouTube, the average number of requests is 4, 3 for three different resolutions and 1 for the still image used as a video cover. It gets even worse when you embed a video that has ads running - several more requests are added. Naturally, your site will load slower and if you have a half a dozen videos on the page ... you do the math, this could be a half of the total requests. What's even worse is that these requests to external resources cannot be controlled, don't have far future expirations and because of that will lower your overall speed scores.
To avoid all of the above, we'll use lightboxes for our videos.
By doing so you will first of all get rid of unnecessary server requests and speed up your site. These requests will be made only when the video is actually played.
The second benefit is a nice crisp video cover with a big button (you'll have to make it) if that's what you prefer in place of a regular video thumbnail. Or simply use a screenshot of your video.
And third, your visitor will not be distracted from your video when it opens in a full screen lightbox.
Your theme may already have a lightbox you could use. Please keep in mind that some lightboxes are just modal popups, requests to the video source are still made on page load.
Some of my favorite lightbox plugins are ARI Fancy Lightbox (you will find the free version in the WP plugin repository) and Modulobox by ThemeOne, found on CodeCanyon.
If you want to make your site blazing fast in under 2 hours, sign up for my full WordPress Speed Secrets video tutorial.