Is shame and guilt the same thing? My initiate thought was 'yes, of course', but after thinking I cannot pin- point the difference.

Shame and guilt share many similarities, they are both secondary emotions and they are both concepts. Being such they are subjective and personally defined by social rules taught (conditioned) during upbringing. There are several competing models for how thoughts and emotions work in the human mind, and a psychologists' description of such is going to rely heavily on which school they are focused on (psychoanalytical, functionalism, behaviorism, humanistic, etc.) So good luck getting a hard and fast answer on this one. However, if one were to use the "window of emotions" theory, which states that emotions can be mapped/charted using two factors: pleasant/unpleasant and mild/intense, then shame and guilt would fall into different spaces. With both being in the unpleasant quadrants but shame being more intense. IMHO

Are these innate feelings, or do they come with socialization, or both?

This is a bit easier, they are both completely socialized.

Are they any use? I mean, can we behave reasonably without them, or are they necessary for keeping us on the right track?

I think they are definitely of use. One could look at shame and guilt as the first line of societal defense. If a person buys into the theory that as humans we have two basic hard wired drives: sex and aggression (one of a few Freudian principles still standing in Psychology today) then shame and guilt are effective in keeping the hedonistic desires of the Id in check. As infants humans are almost all Id, as we develop we move through several phases of conditioning based upon they rules of whatever society we live in, this is where the Superego begins to form and replace the Id.

Or are they barren and useless, or get in the way of functioning well?

I think they can seriously get in the way also. Inferiority complexes are an example of such occurrences.

Interesting topic, thanks for posting it.
~Respectfully,
Tantric