What information does the connectome actually contain? I am not a brain scientist but the following question seems reasonable to me. 

Prof. Sebastian Seung has given a TED talk named "I Am My Connectome". If I understand correctly, connectome is basically the graph whose vertices are neurons and whose edges are synapses (strictly speaking, there are many different kinds of neurons so the vertices of the graph should be colored, but let us forget it for now). 

The title then appears to be a (probably intentional) oversimplification, since even though we do not understand how exactly memories are stored in the brain (and there are many different kinds of memories), things I will roughy call "electrical and chemical variables" play some role in it (so simply remembering whether there is a link or not between two neurons is not going to fly). I understand that studying these "electrical and chemical variables" is like over a half of the brain science but the exact definition is not particularly important for this question. If my understanding is correct, then one should not get overly excited about the present achievements of the Brain Preservation Foundation (the most promising one is, I think, preserving the connectome of a mammal for some time after its death using a chemical solution/freezing), since they have not shown that they can preserve any of the "electrical and chemical variables".

The question is: what information can you actually extract out of a connectome, assuming you do not know anything else about the brain? Like something about a person's health, personality, anything? The chance of them having schizophrenia? Their most likely age?
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