The major difference between butter and oil is that butter is only 80% oil, with the rest being milk solids and water ( source ).
This means that using oil you will lose some of the water content that your cake should have, possibly resulting in a dryer cake.
On the plus side, cakes made with oil tend to dry out slower than those made with butter.
The other possible different, but one that is unlikely with a mix I'd think, is that it depends on the method of mixing in your fat.
If the butter is to be whipped or creamed while softened this creates a different texture than one you could get with oil or melted butter.
Whipped butter or especially butter creamed with sugar provides a network of air bubbles that act as a raising agent during cooking, and the result is a pound cake like crumb.
If your recipe calls for melted butter, though, this is not a concern - the result is more like a traditional muffin or quick bread crumb.
Finally, a vegetable oil has a fairly neutral taste.
Usually when a vegetable oil is called for, this is desirable.
Butter on the other hand has a more interesting flavor.
Your cake may be a bit bland without it.
cooking.stackexchange.com/users/1816/justkt
One big difference is the texture at different temperatures.
Butter is a soft solid at room temp while (most) oil is a liquid.
So the oil-based cake will be moister but also may feel oily.
If you plan to refrigerate and serve cold though, oil can be an advantage because butter is so solid it will make the cake seem tough and dry.
cooking.stackexchange.com/users/1393/Michael Natkin
Believe it or not...
I use yogurt in lieu of the oil.
Works like a champ
and it makes the cake a lot lighter.
cooking.stackexchange.com/users/None/
I have made a box cake with oil instead of butter plenty of times.
I prefer to use oil with strong flavored cakes like chocolate or lemon.
I find that with a yellow or white cake the lack of butter leaves the taste of the cake rather bland.
I supposed if I did use oil instead of butter with a white or yellow cake again I would add a teaspoon of vanilla or almond extract.
cooking.stackexchange.com/users/None/
You can certainly substitute butter with some light oil or margarine, not with high flavour oil like olive oil, though!
The main drawback is the lack of buttery taste of the final result, which is something highly desirable in a lot of cakes.
You can also use yoghurt or lard (in some rare cases).
Yoghurt will make the cake much softer and change the texture.
Lard has a very strong flavour but it is called for in some recipes, like the Italian "Torta Sbrisolona".
cooking.stackexchange.com/users/1537/Sklivvz
Many posts here are saying cakes with oil lack the buttery flavour.
On the co tract alot of blogs on cup cakes and cakes will tell you oil is easier to mix, the taste is also preferable.
You need to use 3/4 the amount of oil to butter when substituting.
I'm not surprised oil tastes better as it has more fat content.
Fat changes the taste texture and odour of food.
Generally fat breaks down chemicals in flavour creating a more concentrated flavour and odour.
I would suggest using baking powder for a more robust cake.
cooking.stackexchange.com/users/79807/Amy Louise Hunt
I want to bake a cake out of baking mix bought at a shop. The mix already contains flour, baking soda, vanillin, other minor components. The manual says I need to add butter.  I'd like to use vegetable oil instead of butter - most likely sunflower seed oil. Is that a good idea? Should I expect any problems? Will the result likely be a decent cake or something to just throw away?
What happens if I use vegetable oil in a cake?
cooking.stackexchange.com
<substitutions><baking><oil>
cooking.stackexchange.com/questions/8828
cooking.stackexchange.com/users/3237/sharptooth
This is possible with a neutral oil, but since butter contains some water you are likely to end up with a moister but more oily result unless you add 3/4 of the amount of oil. You will also lose the buttery taste in the cake. You can also replace the butter with yoghurt for a softer cake. You can replace butter with oil for a box cake. The texture will be slightly different. You should add slightly less oil than in the recipe.
You can certainly replace the butter with oil. It may be a little dryer or even more oily and will lack the buttery taste. Try using yogurt that will make a softer cake. Use less oil than the quantity of butter needed.