My guess is when you look at the animation by dragging the timeline, it looks like you want because you are seeing the poses at full frames, where you have keyed the poses. When you play you will also see interpolation between frames. To set the timeline between frames, hold shift while you are dragging the timeline position. See this section and the Frames section after it:
http://esotericsoftware.com/spine-keys#Timeline-position
You could change your keys to stepped, so you don't get interpolation, or you may be able to animate things differently so you get the look you want from the interpolation.
Note if you are using deform keys, you probably want to avoid them:
https://esotericsoftware.com/spine-keys#Deform-keys
