Check the documentation for underline: by default the stroke and offset depend on the current text settings, so if you use a larger font size for the star, it will affect the underline of that character. As @vmartel08 suggests you can set the stroke and offset values in the underline call to use the same values for the whole piece of text. I would do it like this:
By using em values everywhere we ensure the result will adapt nicely when you change the text size.
The star itself doesn’t look the same because in LaTeX \bigstar is taken from the AMS symbol font instead of the surrounding text font. If you want exactly the same star maybe you can get it from a TrueType/OpenType version of the AMS symbol font like this one.