You can't control each "side" (called handles) of a control point separately because this causes a continuity break in the spline. This may not be important for other software, but for the constraints of how Solidworks works (think about faces), each spline must be continuous.
Thus, to simplify my confusing statement, just right click on the spline, select "Split Entities" (the blue dot on the line), then select the point. This will split one spline into to, and you can adjust the handles seperately. If you still want it to be tangent (but not continuous), just click on both and add a tangent constraint.
As for moving one point and having it affect others, if you click on the spline (not a point, but the line), is the "proportional" option turned on? Turn it off. If it still does not behave as you expect are there any constraints in your drawing that are effecting the points? Typically this only happens to me when there are extraneous contraints. Try drawing your spline on a brand new sketch with nothing else.
Actually what I think you meant to say is "When I move one control point the other control points don't move but the shape (vectors and tangent weighting) of the entire spline changes." The exception to this is when you are using the control Polygon for editing.
If my statement accurately reflects your experience, the main difference between SW and other packages (like Alias and Rhino) are that our splines are continuously "parametrized" - that is as you edit the spline the entire spline's parameters are being recalculated automatically. We do this so that splines can somewhat predictably update when parent references move. A way to get the behavior you desire is to dimension the vectors and tangent weight of the spline points that you do not desire to move - this will greatly minimize (but not totally eliminate) movement thruout the rest of the spline shape.
None-the-less, there are possibly more elegant ways we could give you the behavior of "other" CAD packages - I would encourage you to please submit an enhancement request here
Which "other software" are you comparing to? After reading Mark's comments, I can see we thought differentely about what you are talking about. I was comparing with 2D drawing software, like Adobe Illustrator.
In Illustrator, you can drag each side of the handle seperately after pressing a key (CTRL?).