No, IE and FireFox don't render ­&lt;/br> the same as <br />. They render </br> as if it were <br /> (and even render <br></br> as if it were <br /><br />). If you want to use PerlMonks to test that, then turn off HTML nesting enforcement so that the browser will see </br> (not &lt;/br> or nothing).

HTML nesting enforcement will remove and/or add tags to attempt to make the node well-formed so it won't impact the display of surrounding material. If you have its error reporting turned up high enough, then it will also add greyed-out information about what was removed or added. So the original (invalid) </br> is removed by it and, for you, a greyed-out version is inserted so the transformation is not "silent" in this case.

It is sad that "view source" (in the case FireFox, at least, last I checked) doesn't always faithfully show the source that it received so that people are left guessing.