Why is schizophrenia a thought disorder and bipolar a mood disorder? Examining two major mental illnesses that are,  loosely, grouped by mood and thought.

Schizophrenia is regarded as an illness associated with thought disorder. 

Bipolar is regarded as a mood disorder.

Both conditions can lead to a psychotic break and behavioral disturbance. Hallucinations can occur in both. 

Bipolar is characterized by depression with the occurrence of, at least, one episode of mania. 

Schizophrenia is characterized by phases. An active phase, where psychotic symptoms peak and the negative phase, when the personality has little affect. 

Given that both cause behavioral problems and changes to mood. That psychosis is based on false beliefs, and beliefs are thoughts. What is the scientific and the psychiatric basis for regarding schizophrenia as a thought disorder and bipolar a mood disorder?
In this paper, the authors note that bipolar disorder and schizophrenia share many abnormal resting state network connections. But some connections are specific to bipolar disorder and others are specific to schizophrenia. 

I think to really describe the two illnesses, one must look at specific biological data. Fuzzy terms like "mood disorder" and "thought disorder" are used without any concrete meaning. If somebody looked at the fmri data he would be able to tell the difference between bipolar disorder and schizophrenia (without knowing any psychiatric terms). At the same time, I think looking solely at fmri data can be too abstract without being informed by psychology and psychiatry. A good book on this subject is Brainwashed: The Seductive Appeal of Mindless Neuroscience.

There is also some evidence that time perception may be altered in various mental illnesses. So one may distinguish bipolar disorder and schizophrenia in terms of differences of time perception.

Source

Differences in resting-state functional magnetic resonance imaging functional network connectivity between schizophrenia and psychotic bipolar probands and their unaffected first-degree relatives.
Meda SA, Gill A, Stevens MC, Lorenzoni RP, Glahn DC, Calhoun VD, Sweeney JA, Tamminga CA, Keshavan MS, Thaker G, Pearlson GD.