Have you taken a look at wikipedia page for Scalar curvature? [BTW, always a great resource!] There you can find the standard geometric interpretation of Scalar curvature, as measuring the volume distortion on balls of small radius, compared to Euclidean balls of such radius. Analogously, Ricci curvature in a direction measures volume distortion of small ``rods'' along that direction (and then these interpretations make sense together in terms of scalar curvature being an average of Ricci in all directions). I think that's possibly the most geometric picture you can get (apart from interpretations from mathematical general relativity).
As for what "scalar curvature forgets, but Ricci still sees" you can think in terms of topological obstructions for these curvatures to have a certain sign; e.g., manifolds with positive Ricci curvature (bounded from below) must be compact (you even have an estimate for their diameter) and have finite fundamental group. Instead, manifolds with positive scalar curvature can be much wilder (in particular, they might not have finite fundamental group). Also, you can look at the problem of prescribing Ricci vs. prescribing scalar curvature, one is clearly way more flexible than the other, see this post regarding Kazdan-Warner's stuff for scalar curvature and compare with obstructions to positive (and non-negative Ricci curvature). Although endowing a given manifold with a metric with positive/nonnegative ricci may be impossible, it certainly always has tons of metrics with negative ricci curvature (see this paper), in that such metrics are actually $C^0$-dense in the space of all metrics.