Both are supported in HTML 5, so let's take a look at them in turn:
The <embed> tag
To embed a file into your page, the syntax is simply:
To pass a parameter into the plugin used to display the content, you can simply add it as an attribute:
The object tag is newer than the embed tag and so has a few more features.
The basic syntax for embedding a Flash file is:
To pass parameters to the plugin, you don't use an attribute, you specify a <param> tag within the object tag itself, like this:
The object tag will also allow you to embed another webpage, but this is discouraged as the <iframe> tag is designed for that purpose.
One of the benefits of using the object tag is that if the browser doesn't support the content that you are embedding, you can add fall-back content within the object tag, which the browser will use instead:
Which one should you use?
The object tag has a number of advantages, the most important being the ability to add fallback content. From what I can see, the only reason you would use embed is to support older browsers, (IE6 and earlier).