One is a function (ie. a scalar field, an expression) and the other is an equation, so they are fundamentally different. However when you set the stream function equal to a constant scalar you will get an equation which defines a streamline. So they are pretty much identical, physically. So the stream function is constant over streamlines, but they are not technically the same- one is a field, the other is a family of curves.
Streaklines are the path a fluid drop will follow in the flow. It is a little confusing, but think of a river which gets dammed. Before it is dammed, the streamlines are straight downstream, after it is dammed, they all change because they must pass through a small hole. However, think of the streakline of a fluid particle whic is released way upstream- if the dam gets built before the particle reaches it, that streakline will pass through the hole in the dam- it will not change because it is not a function of time, only initial conditions.
So an analogy would be streamlines representing the direction of a gravitational field, and streaklines representing where a mass actually goes in that field. Although the gravitational field in the solar system is always changing due to planetary motion, the streakline of, say, the Voyager probe will be only dependant on the initial conditions (and so streaklines require knowledge of the streamlines for all time). If the flow is steady, they become streamlines.
Bottom line: streamlines are lines in space depending on time. streaklines are lines in space depending only on an initial time and position, after that its path is fully described by the flow field.