Oils-Pros: bright, vibrant colors that you can mix into any color under the sun
you can add thinkeners, thinners, impastos, and other mediums to really achieve the look you want. They take a loooong time to dry which means you can really play around with the paint and don't have to worry about making a mistake. This also makes blending a lot easier. You can use the differences in the transparent vs. opaque paints to create different layers which can add richness and depth. Overall, I just like the feel of oils...it's hard to describe.
Cons: they are more expensive than acrylics, I spent about $150 just to get a set of basic student level paints and brushes. They are toxic, you should work in a well ventilated area (i am not good at doing this but I think my brain is still good, haha). They are more difficult to clean up, you need to use a paint remover like turpentine, white spirit, etc to clean the brushes and yourself. This can be a bit of a pain. They take a loooong time to dry (pro and con) some people get annoyed because you have to let the piece dry before you can work on it again without blending the color.
Acrylics -
Pro: also bright, vibrant color and the option to add a thickener but it's more likely to just use the paint straight from the tube. Dries VERY quickly so you can work fast and finish a whole painting quite quickly. You can also layer but these paints are mostly opaque. Relatively cheaper, water soluble so cleanup is very easy, just rinse in some water.
Con: They dry too quickly! Even mixing colors on a palette is harder because you have to mix the right amount for the whole painting once or it will harden and you'll have to mix all over again making sure to try and match the color! You also have to keep you brushes wet while working, if it dries, your brush is finished. You can't really achieve the deepness and richness of color that you can with oils.
And the debate will continue!