deformation-retracts into a point / contractible : what is the difference?

For a contractible space, the homotopy is allowed to move the point, we don't need to have

$$h(p,t) = p \text{ for all } t \in [0,1].$$

For the (strong⁽¹⁾) deformation retract, $p$ must be kept fixed by the homotopy.

So (strong) deformation-retractability is a stronger condition.

⁽¹⁾ Different authors use different terminology. Some call simply deformation retract what others call strong deformation retract, and that seems to be the case if "spaces that deformation retract onto a point are contractible, but the opposite is not necessarily true". If a deformation retract need not keep the retract fixed in the homotopy, the two properties are exactly the same.