What is the difference between 款式 and 格式
The way out of your conundrum is to realize your original assumption I know both of them mean "style" is the source of confusion.

Words can become flexible and have more than one meaning, or "senses of the word". However, usually when asking about the meaning of a word, we are talking about the primary, or most common usage of a word.

In this case, even if both 款式 and 格式 can sometimes be used as "style", only one of them can be considered "style" as its primary use, primary meaning.

款式

Out of the choices, only this would have a primary meaning of "style"

CDICT search for 款式 translates this as:


pattern
style
design


格式

CDICT search for 格式 translates this as:


form
specification
format


On a more technical note, Yellowbridge etymology on the word 格 shows


木 [mù] tree; wood, lumber; wooden
各 [gè] each, individually, every, all


Perhaps one way think of what this word means, is that, at least in ancient times when this word was developed, wood was commonly thought of as a practical material for making, reshaping all kinds of different "forms" or "things"

In Taoism, Traditional Chinese Medicine, Feng Shui, within their five elements, 木 or "wood element" is associated with creation/creating.

Conclusion

So, as long as you can let go of the idea that both "mean" style, then you can see that the difference in these terms is:


款式's primary use/meaning/sense is "style"
格式's primary use/meaning/sense is "form"


Unless you are not sure what the difference between the words "style" and "form", that would be new question then.