Why is schizophrenia 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.