What information does the connectome actually contain?

Well, regarding this short definition of a connectome from Wiki:

A connectome (/kəˈnɛktoʊm/) is a comprehensive map of neural connections in the brain, and may be thought of as its "wiring diagram". More broadly, a connectome would include the mapping of all neural connections within an organism's nervous system.

Therefore you could read out all information, assuming you have enough computational power. (A perfect connectome would represent all information processes in the brain. Because processing information through connections is all the brain does, right? :D)

In the real world however, we need 'models' as simplifications to deal with these phenomena. A connectome is nothing else than such a model. The classical model as vertices and nodes approach (which leads to solving differential equations in the end) is mainly based on the mathematical field of graph theory and probabilistics. Also measuring brain properties has always to do with oscillations and signal theory. (Note on simplification and prediction: The fact that a model is simplifying should be independent of the quality of prediction of outcomes. The model might me accurate, even if it is a simple one.)

Some thoughts on the limits of obtainable information

1. To make predictions of real-world constructs, you need to assess real-world phenomena. F.i. you need to have postulated some outcome-variables (for lets say Schizophrenia) that need to be quantified in the real-world (external validity) to be related to inner-brain processes. These real-world concepts of psychological processes, speaking of the behavioral, emotional and cognitive aspects (which can be measured), are inherently fuzzy, which raises the question, to which certainty we will be able to predict psycho-physiological correlates.

2. Not to forget, memories are mainly coded as 'processes over-time', something like chains of activation (other than distinct anatomy and brain-areas who would define thoughts). Also noisyness and randomness is always a variable you need to consider. (I think the Hippocampus functioning of neurons induces randomness by design, but I have no source right now.)

3. Also to mention: The individuality of brains leads to decreased prediction of the model or a general model that would hold for every individual

4. The plasticity of the brain states states the ability to change and evolve over time. (Concepts like Imprinting and so on). A static model might also not be sufficient to represent these (also highly individualized) processes.

5. Situational- or environmental dependency also influences your current state of mind. (It is not possible to 'control' all environmental factors, unless you had infinite computational power.)

6. Also the brain and toughts are highly self-/auto-activating, recurring and in parallel. Millions of neurons firing at the same time and oscillating in rhythms. Thoughts that become conscious might only be the tip of the iceberg in a sea of thoughts that are below a certain threshold.

My proposal to your question: "What exactly can you get from a connectome?"

In principle everything is possible, but practically we might never truely decode everything from a philosopical and mathematical point-of-view (we needed to create a perfect connectome, which would be a perfect simulation of the entire brain). We might however be able to find very accurate brain-markers that help in predicting real-world concepts that seem relevant for humans :)

Resources

Because it is great: Video of synchronizing fireflies

Buszaki: Rhythms of the Brain is a nice read on some how information is created in the brain and different processes