I'm not so sure about what you mean by 'save'. Is using the submit button to save the grid a documented feature in ColdFusion? I suspect not. If indeed not, then any results you obtained with it were just accidental.
Some things we know for sure. A submit action brings you back to the form page. However, you are dealing with 2 different page contexts.
After the submit button is clicked, the form page currently before you is the action page. It has a different context from the submitting page, which is now the referrer. The cfform tag has an attribute, preservedata, to maintain field values after submission. Alas, the documentation says that this doesn't apply to grids.
I have always thought the action to save records after an update in cfgrid is to "press the ENTER key on the keyboard". Look it up.
If that is so, then you might want to try something like
<cfinput type="button" name="SaveChanges" value="Save Changes" onclick="return String.fromCharCode(13);">
Here, onclick="return String.fromCharCode(13);" is my attempt to model "press the ENTER key on the keyboard". It might be wrong. Just an idea to point the direction.