Something that's powerful displays, well, power. Your neighbor who just finished training to be a fireman probably looks much more powerful than he used to.
We use powerful for many different senses of "strong." A powerful medicine is very effective. A powerful politician has more influence than most politicians. A powerful dog could probably knock you down. If you're reading a book written in folksy Southern vernacular, someone might use powerful to mean "very," as in, "that was a powerful good supper, Miss Jackson," or "you look powerful good tonight."