No, I don't have another template - I know what I 'm talking about and this is about the Beez that comes with Joomla 3.
I replaced the image with another of the same size and it's not looking bigger or isn't fitting. I am getting the exact same behavior as if I was using the personal image.
The description of your problem is rather vague for us to understand what the actual problem is. Maybe what you are trying to do is not going to work.
A few things to note about Beez:
Beez actually has a certain way of dealing with the header image in its css - that positions the background image to top-right.
It doesn't provide css that will resize the background image on window resizing, - not exactly what you would call "responsive" at this element.
Also another thing to note is that the personal.png background image has bigger height than the actual header area height that it covers.
To be frank, using the Beez template / personal style and try to replace it's background image for the header, it is not the most easier/user-friendly thing for a beginner user, as it requires from someone to understand how its css works, or just simply use a header image that is going to work in the exact same way as the default one -and this is not only about using the exact image size.
Set back again the original image and study how it behaves.