For the Soft Body to follow a path you need to have the Soft Body Goal enabled. The Soft Body Goal can either be defined for all vertices (if you don't specify a Vertex Group) or for specific vertices - with associated weights - as a Vertex Group.
Think of the Soft Body objects as a combination of two meshes. One is the 'actual' mesh - and this can be animated - if you disable the Softbody modifier in the modifier stack then the mesh will revert to that form.
The other 'mesh' is the Soft Body mesh. This is separate from the 'actual' mesh and its vertices are moved by the Soft Body simulation. Any simulated vertices flagged as being a 'goal' vertex will attempt to move towards the corresponding 'actual' vertex - with the strength determined by the Goal Strength and the Vertex Group Weight. If you set the weight to '1' then the simulated vertex will always exactly match the corresponding 'actual' vertex whereas lower weights will experience a smaller pull towards that 'goal' location.
Since you're seeing the Soft Body fall under the pull of gravity that indicates that you are not using the Goal - so despite you animating the 'actual' mesh, the simulated soft body vertices will not be 'pulled' towards it - so it ignores your animation.
So get it to follow the animation simply enable the 'Goal' and set a suitable Goal Strength to follow your path while still allowing it some sift body "wobble" - or use the Vertex Group so that only some of the vertices have a 'goal' and the rest will be pulled along with them following the soft body simulation.