Note: the difference is [edit] very small [/edit], so zooming may be necessary to clearly observe the difference.
Not sure about 4px difference. Box model inspector says btn height is 35px, while form-control is 34px.

I have the same issue:
<div class="input-group"><input data-arp="false" name="dateOfBirth" type="hidden"><input id="dateOfBirth" class="form-control" type="text" data-validation="true" data-optionality="required" data-required-message="Value is required."><span class="input-group-btn"><button title="Show Calendar" type="button" class="btn btn-default"><span class="glyphicon glyphicon-calendar"></span></button></span></div>
yields:

But adding some text to the button:
<div class="input-group"><input data-arp="false" name="dateOfBirth" type="hidden"><input id="dateOfBirth" class="form-control" type="text" data-validation="true" data-optionality="required" data-required-message="Value is required."><span class="input-group-btn"><button title="Show Calendar" type="button" class="btn btn-default"><span class="glyphicon glyphicon-calendar"></span> Button
</button></span></div>
yields:

Which is what I want ... without the extra text. For the moment I'm using a .
@hlship did you ever figure out why your buttons were doing that? I'm seeing the same behavior as your images in chrome and I don't see it in the bootply url above with the same markup I have locally. I set the line height of the glyphicon to inherit instead of 1 and it "fixed" it for me but thats obviously a bad solution.
I just (temporarily) added a which fixes the element's height correctly.
Closing for #10936.
The cause is "" first string. It must be set as writter above for html 5. If it doesn't exists or it has html 4 description ("PUBLIC etc") this error occurs.
Adding solved the problem for me. Thank you katrazka!
Most helpful comment
The cause is "" first string. It must be set as writter above for html 5. If it doesn't exists or it has html 4 description ("PUBLIC etc") this error occurs.