Cold butter mixes with flour and creates pockest of fat that melt and make for a lighter more airy pastry or crust. I have never seen a recipe for a crumb topping calling for melted butter.
Melted butter acts more like oil and is more often used for cooked frostings or mixed with melted chocolate.
It would help if you could give specific examples.
It's not a matter of chemistry but of physics.
Linda C

What Linda said... solid VS liquid and physics.

When making a pastry, for instance, you cut cold solid fat (butter, lard, or shortening) into the flour. Wherever the fat coats the flour, the gluten won't form when water (or other liquid) is added to the mix (liquid activates the gluten). Because the fat is mixed throughout the flour mixture, when the little globs of fat melt during baking, the steam is released and the layers of pastry separate where this occurs, and that's how you get a flaky pastry.

To assure both tender and flaky pastry, try this technique. Cut half the fat into the flour until it's a fine mixture. Leave the other half of the fat in somewhat larger chunks and add it after the addition of the first fat. Don't over-mix. The first addition of fat yields tenderness, the second addition of fat yields flakiness (marked by the "blisters" on a flaky pastry).

When you make pastry using oil (or a melted form of a solid fat), it coats each particle of flour so thoroughly that it completely prevents the development of gluten in the flour. Using a liquid fat will make a very tender crust, but lack those "blisters", and liquid fat in pastry tends be mealy, rather than flaky, and are more difficult to roll out due to it's tenderness.

You will also get different results when using oil in a cake recipe, instead of the traditional (shortened) cake method where the solid fat and the sugar are creamed, which adds volume from air. Unshortened cakes rely on foam formation for their structure - angel food sponge and chiffon cakes are unshortened cakes. Chiffon cakes (the hybrid of of shortened and unshortened cakes) are more tender than either angel food or sponge cakes because of added vegetable oil.

"In traditional muffin making, all the steps are geared to minimize mixing. The fat is most often in liquid form, either melted shortening or an oil. It's usually mixed with eggs and the other liquids, then added all at once to the dry ingredients. Since fat affects tenderness, muffins with a high fat content not only taste rich they also have a softer crumb." (Source for quote: The Inquisitive Cook"). If you have a muffin recipe that uses traditional cake-making techniques, the solid fat is creamed with the sugar and the muffin will have a more cake-like texture.

Those are a few of the distinctive characteristics between the same food made with a liquid or solid fat.

-Grainlady

Also, if a recipe calls for 'X amount butter, melted', it means you should measure out the butter while it's cold, then melt it. If it calls for 'X amount melted butter', melt the butter first, then measure. It can make a big difference!

Grainlady : I knew that !!!

And what Linda said.

Lou

I've seen melted butter used in a crumb toppings for savory dishes, e.g. stuffed tomato, casseroles with chicken or fish, macaroni and cheese. Why is melted the preferred choice here?