Well, the first obvious difference would be that the square-cube law would basically be replaced by a cube-tesseract* law. Since 4/3 is closer to 1 than 3/2, this means that there could be greater variations in size. Another difference is that more of the (hyper-)volume is close to the (hyper-)surface. Since this means that important organs are closer to the surface, probably organisms would grow thicker protective skins. On the other hand, plants would probably profit from the increased surface/volume ratio, as they could keep relatively compact form and yet present a large surface to the light. Therefore I guess leaf structures would be less common, or otherwise, plants developing those would have more energy at their disposal, allowing them to evolve some more sophisticated abilities.
Since the floor would be three-dimensional, you'd need at least four legs on the floor for a stable standing (instead of three, as in our three-dimensional world). Therefore for an insect-like moving pattern, you'd need 8 legs. Coincidentally, if you extrapolate the four-legged pattern of the land vertebrates to four dimensions, you also get 8 legs. Therefore I conjecture that eight-legged animals would be common in four dimensions. Probably the typical four-legged animal would be long on one horizontal axis, and when looked on from above (so you get a 3D projection, similar to the 2D projection when looking at out animals from above), you'd basically see an ellipsoid with four left attached close to the front, and four close to the back, each arranged in a square. A stable standing would be achieved with two diagonally opposed front legs, and the two back legs on the other diagonal. From that fact, possible walking patterns could be derived.
When moving to an upright position (similar to humans), those creatures would walk on their four hind legs (interestingly, unlike our two foots in 3D, those four legs would still be enough to provide a stable stand without the help of specially formed feet, if positioned right), and have four arms available for manipulating things.
*) A tesseract is the four-dimensional equivalent to a cube.