There is evidence that people think and act as if there are different kinds of unknowns, and regard them as having different consequences.  The best evidence comes in the form of behavioral experiments and brain-imaging studies.

I'll focus here on the debates about whether uncertainty all boils down to probability. These debates go a long way back, but in modern times the classic demonstration that there may be more than one kind of uncertainty is Ellsberg's (1961) distinction between ambiguity and probability.  Building on his work, I provided experimental evidence that people distinguish ambiguity from conflict or disagreement, and tend to regard conflict as a worse kind of uncertainty than ambiguity (Smithson, 1999).  This yielded a productive line of research that has continued on for more than two decades (I provide an overview in this chapter).

For a gentle introduction to these concepts, see my blog post. A fun example of the debates about whether the distinction between ambiguity and probability makes sense is provided in this blog post. For a more technical treatment and recent experimental work on how people perceive conflicting versus ambiguous information, there is my 2013 conference paper. Arguments and examples of the practical importance of this distinction can be found in yet another blog post. And for an overview of how humans (mis)understand probability, see my chapter in the Oxford Handbook of Probability and Philosophy (edited by Alan Hajek and Chris Hitchcock).

Another important distinction is between uncertainty about the probability of an alternative and a lack of knowledge about which alternatives are possible. Sometimes this is referred to as state space ignorance or sample space ignorance.  Because probability judgments depend on how the state space is partitioned (e.g., whether we believe there are 2, 4, or 10 alternatives), an unknown or indeterminate partition poses difficulties for such judgments. I initially did some experimental work on this topic with one of my Honours students and a Japaneses colleague (Smithson, Bartos, & Takemura, 2000). Several years later I wrote papers on how ideas about state space ignorance pose problems for probability judgments and whether there are "rational" guidelines to the number of alternatives we should want (Smithson, 2006 and 2009). One of my Honours students and I also demonstrated experimentally that partitions are problematic even if we allow probabilities to be vague (Smithson & Segale, 2009). Two more of my Honours students and I applied these ideas to the debates about the merits of the Scottish "Not Proven" middle option for verdicts in courts of law (Smithson, Gracik & Deady 2007). 

And I've been involved in brain imaging studies with Helen Pushkarskaya and her colleagues.  We have a paper out demonstrating that processing sample space ignorance entrains different structures in the brain from those engtrained when processing ambiguous or probabilistic uncertainty, although some structures are shared across these three kinds of uncertainty (Pushkarskaya et al. 2010), and another paper (Pushkarskaya et al., 2015) on how the brain deals with conflicting versus ambiguous information.

References
Pushkarskaya, H., Liu, X., Smithson, M. and Joseph, J.E. (2010). Beyond risk and ambiguity: deciding under ignorance. Cognitive, Affective, and Behavioral Neuroscience, 10 (3), 382-391.

Pushkarskaya, H., Smithson, M., Joseph, J.E., Corbly, C., & Levy, I. (2015) Neural correlates of decision-making under ambiguity and conflict. Frontiers in Behavioral Neuroscience, 9, 325.  doi: 10.3389/fnbeh.2015.00325..
Smithson, M. (1999). Conflict Aversion: Preference for Ambiguity vs. Conflict in Sources and Evidence. Organizational Behavior and Human Decision Processes, 79: 179-198.
Smithson, M., Bartos, T.G., and Takemura, K. (2000). Human judgment under sample space ignorance. Risk Decision and Policy, 5, 135-150.
Smithson, M. (2006). Scale construction from a decisional viewpoint. Minds and Machines 16, 339-364.
Smithson, M., Gracik, L. & Deady, S. (2007). Guilty, not guilty, or … ?  Multiple verdict options in jury verdict choices.  Journal of Behavioral Decision Making, 20, 481-498.
Smithson, M. & Segale, C. (2009) Partition priming in judgments of imprecise probabilities. Journal of Statistical Theory and Practice ,3, 169-182.
Smithson, M. (2009) How many alternatives? Partitions pose problems for predictions and diagnoses. Social Epistemology, 23, 347-360.