I am struggling to find the difference between the two concepts. From what I understand both refer to turning raw data into more comprehensive features to describe the problem at hand. Are they the same thing? If not could anyone please provide examples for both?
Feature extraction is usually used when the original data was very different. In particular when you could not have used the raw data.
E.g. original data were images. You extract the redness value, or a description of the shape of an object in the image. It's lossy, but at least you get some result now.
Feature engineering is the careful preprocessing into more meaningful features, even if you could have used the old data.
E.g. instead of using variables x, y, z you decide to use log(x)-sqrt(y)*z instead, because your engineering knowledge tells you that this derived quantity is more meaningful to solve your problem. You get better results than without.
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