I recently wrote this to a friend of mine and it seems relevant, so I hope he won't mind that I share it here:
Going from anxiety/OCD to schizophrenia can seem like a "logical jump" in our own minds because the intrusiveness of the thoughts that characterize those thought patterns. However, schizophrenia and the anxiety disorders are on totally different spectrums from one another: while, admittedly, schizophrenia could certainly lead to anxious thoughts, it doesn't work both ways. Anxiety, OCD, PTSD and the like does not and cannot morph into something like schizophrenia. (Arguably, traumatic events such as drug use, and acute stress can trigger the onset of schizophrenia, but in those cases, it generally would have happened at some point because the components were there from birth.) I believe that it's now widely accepted that schizophrenia is largely genetic in nature, due to both hereditary factors, brain chemistry and the structure of the brain itself.
Anyways, the onset of schizophrenia is, in the vast majority (if not all) of the cases, not recognized by the affected individual themselves. This is purely due to the intrinsic nature of the disorder and what can characterize anxious "intrusive thoughts" from psychosis: the individual is unable to see that the thoughts are irrational. For example, John Nash never questioned his delusions that aliens were talking to him because the voices "came to [him] as the mathematical ideas did." That's where the whole "if you were going crazy, you wouldn't be asking if you're going crazy" thing came from. Schizophrenia is generally noticed by people close to the individual when they see that something very wrong is going on.
So there is a definite difference between true delusions and intrusive thoughts, even though it seems like the two are closely related.