It's most likely the color of the PNG image that is not displayed consistently.
A PNG image doesn't have color space information, instead it has a gamma value, and there is a problem to interpret that value to determine a color space. You will probably see that there is a color difference between different browsers, so if you adjust the color for how one browser displays the PNG, it won't match in other browsers.
Use a different file format if you need the color match other elements, or make the background of the PNG transparent instead of green.
Rendering text is a different matter. There will always be slight differences in how browsers renders different fonts, depending on the rendering method used, which fonts are installed, and system/user settings. You simply can't expect exactly the same result in different browsers.