Why would someone seek treatment for Apraxia of speech? People with Apraxia of speech have two basic problems: monotone pronunciation and slow speech.  

(1) Might someone with Apraxia of speech who does not seek treatment have psychological problems? They just don't care about not talking with people who don't want to wait and listen to their thoughts. Is that arrogance? low self-esteem? severely anti-social?

(2) Is Apraxia of speech degenerative? Without treatment, speaking will become more, and more, difficult?

(3) Is it socially unacceptable to speak slowly with a monotone pronunciation?
There appear to be more than your two basic problems: see Wikipedia's list of characteristics. Subjective difficulty in producing speech appears to be one particularly plausible reason to self-initiate treatment.


I don't see any indication of a lack of concern for how speech is received by others.
Some forms seem to be progressive, but not all are:


  Recent research has established the existence of primary progressive apraxia of speech caused by neuroanatomic motor atrophy.[19][20]


By comparison, stroke can introduce acute apraxia of speech that can (rarely) improve over time.
I wouldn't say "unacceptable", but undesirable seems true enough in general. Plenty of variation in rate of speech is normal and acceptable, but extremely slow speech can be grating. Tonal variation is expected in most languages with which I'm at all familiar, but again, most people probably understand that some people will be more or less monotone. Again, extreme monotonicity will tend to violate expectations and raise complaints, because variations in pitch and emphasis carry useful semantic information in normal speech. Personally, I've been criticized for speaking in monotone at many times in my life, but never for having too wildly fluctuating intonation, even though I've tried to exaggerate this quality somewhat compensatorily. Yet as with all anecdotes, your mileage may vary.




References
19. Josephs, K. A., &amp; Duffy, J. R. (2008). Apraxia of speech and nonfluent aphasia: A new clinical marker for corticobasal degeneration and progressive supranuclear palsy. Current Opinion in Neurology, 21(6), 688–692.
20. Josephs, K. A., Duffy, J. R., Strand, E. A., Machulda, M. M., Senjem, M. L., Master, A. V., Lowe, V. J., Jack, C. R. Jr., &amp; Whitwell, J. L. (2012). Characterizing a neurodegenerative syndrome: Primary progressive apraxia of speech. Brain, 135(5), 1522–1536. Retrieved from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3338923/.