In regular conversation there is no difference and they can be used interchangeably. You are just implying that something isn’t very far, or that there are not turns. “Where does Jon’s house?” “Oh, his house is just up/down the road from here.” If there is literally an incline or decline, like a mountain road, then maybe you would be more specific with up or down. But in general, it isn’t a phrase that is used to give an exact direction, just to imply how close or simple to travel to a place is.

There may be no difference unless it's obvious which way is uphill and which way is downhill - "up the road" implies uphill and "down the road" implies downhill. In particular contexts, people also use "up" in connection with going / being to the north, and "down" in connection with going / being to the south, but when you're talking about a short distance on the same road, that would be too complicated. In very specific contexts, people also might use "up" meaning "to the university" and "down" meaning "from the university" but that might be specific to particular universities.