Potentially better usable high ISO range (when comparing like for like bodies of similar ages the larger sensors always win over the smaller ones)
Greater degree of background blurring - about one stops worth when comparing like for like shots (generalist comment, its a bit more complicated than that, but essentailly bigger sensors means more background blurring)
Wider angle of view on focal lengths compared to the same focal lengths on crop sensor cameras.
This latter point is often the biggest one people talk about - mostly because for decades the 35mm film (which is the same size as "fullframe digital) was one of THE most popular film sizes in the casual and pro markets. As a result when crop sensor digital hit the market you got all these "its just like 1.5 times the normal range" normal being compared to 35mm. Of course if you've never shot 35mm/fullframe then you've no comparative experience to draw from so you're more going at it the other way since what you've seen on the crop sensor is your view of "normal"
That said for indoor portrait photography and similar situations the wider angle of view helps since it allows a photographer to use a telephoto instead of wide angle lens for many shots. This helps avoid perspective distortion (bits closer to the camera get enlarged over those further away) and is why the larger sensors are still popular today.