Jack ate the last cookie if only because he wanted to wash the plate.
Jack ate the last cookie because he wanted to wash the plate.
The difference is that the second sentence leaves out the idea that the
only
reason for eating the cookie was to wash the plate. It also
leaves out that certain stylistic 'something' that the first has.
The reason in the first sentence has a certain subtractive value,
whereas the reason in the second has additive value. In the first
sentence,
but only because might replace
if only because. In the second sentence, one can easily imagine extending the thought with further reasons:
because ... and because ... and because ...
CJ