The still bunny is nice but the animated bunny has a few beginner mistakes. Overall it looks like the work of a timid animator, avoiding any large motions. The result is very stiff and lifeless. I have two pieces of advice.
1) Give yourself more room to work with when animating; your bunny looks like he is wedged into a box and it's not giving him room to move realistically. Use a much bigger canvas with plenty of empty space on the sides and top, and then crop it when you're done. This will help avoid claustrophobic, timid movements. Stretch his limbs out when he hops, don't be shy!
2) Try working with keyframes. This is an old, kind of lame example of mine:
http://www.wayofthepixel.net/pixelation/index.php?topic=4599.msg57614#msg57614 Draw the extreme positions first and then fill in the frames between. Right now you're trying to make the whole animation with one frame that you slightly modify. This will
always yield poor results without exception, it's just too restricting for an animator to create compelling motion. If you want a nice animation you're going to have to draw the character more than once. This is true for pixel art, for flash animation, for theatrical animation, and everything in-between.