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 &quot;wiring
diagram&quot;. 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.
Influential Factors to consider

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.

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.)

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

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.

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.)

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: &quot;What exactly can you get from a connectome?&quot;
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