The difference is that $\sin(x)$ "starts at" $(0,0)$ and $\cos(x)$ "starts at" $(0,1)$ (note: By "starts at", what we really mean is "the y-intercept is at").
What you are doing is graphing sines and cosines by taking into account elementary transformations - horizontal/vertical shifts/dilations, etc. and seeing where this point ends up under these transformations.
In the case of $y = a + b\sin(c(x - d))$, we have a horizontal shift right $d$ units, a vertical dilation of $b$ units, and then a vertical shift up $a$ units. So, if we track where the y-intercept goes under these transformations, it goes from $(0,0)$ to $(d,0)$ after the horizontal shift, the dilation does nothing to this point (since $0*b = 0$!), and then finally to $(d,a)$ after we do the vertical shift.
For a $\textit{cosine}$ however, the original y-intercept is $(0,1)$, so, for $y = a + b \cos(c(x-d))$, the horizontal shift takes us to $(d,1)$, the dilation to $(d,b)$, and the vertical shift to $(d,b+a)$.