we talk in General Relativity [...] about curvature of space and time
At least (without implying strict separability) about curvature of spacetime; right.
In this context it is however possible to distinguish "free (inertial, falling) motion" from "any other motion", The latter is accordingly said to be "due to (external) influence(s)", and its ("first order") deviation from "free motion" is quantified as a non-zero value of (instantaneous) acceleration.
Is there a principal difference [...] ?
The difference is most plainly and strikingly that the acceleration value is either zero, or non-zero.
A general prescription how to evaluate acceleration (from given more basic geometric notions, namely given values of spacetime interval ratios) is sketched here (PSE/a/146397).
Note that this mentioned prescription is based on non-zero spacetime intervals; it is indeed intended to be evaluated for ratios of certain time-like intervals. It is not even defined for a set of events which are all (pairwise) separated by null intervals. However, it may be argued that it is therefore consistent to attribute to the propagation of a signal front strictly an acceleration value of zero; calling it strictly "free" with respect to the possibly curved spacetime geometry of the experimental region under consideration.