What is (or is there) a clear distinction between cognitive and motivational explanations for a hypothesis? For example, I see there is even a cognitive theory of motivation. I closest I can find is that cognitive explanations are more about the thought process of an individual which led to certain behaviour. What then, does a pure "motivational explanation" mean?

Here is an instance of cognitive vs. motivational explanation: "There are both cognitive and motivational explanations for the competence hypothesis. [...] Perhaps the major reason for the competence hypothesis is motivational rather than cognitive." This is an excerpt from "Preference and Belief: Ambiguity and Competence in Choice under Uncertainty" by  Heath and Tversky.
Summary: it appears that the contrast from the paper cited is based on a 1970s paradigm where a cognitive approach meant information processing applied only to problem data, whereas the motivational approach allowed for other variables, such as self-esteem, which nowadays would be described as an affective variable.



Here's the fuller context for the claim:


  we propose that—holding judged probability constant—people
  prefer to bet in a context where they consider themselves knowledgeable or competent
  than in a context where they feel ignorant or uninformed. We assume that our
  feeling of competence
  in a given context is determined by what we know relative towhat
  can be known. Thus, it is enhanced by general knowledge, familiarity, and experience,
  and is diminished, for example, by calling attention to relevant information that is not
  available to the decision maker, especially if it is available to others.
  
  There are both cognitive and motivational explanations for the competence hypothesis.
  People may have learned from lifelong experience that they generally do better in
  situations they understand than in situations where they have less knowledge. This expectation
  may cariy over to situations where the chances of winning are no longer higher
  in the familiar than in the unfamiliar context. Perhaps the major reason for the competence
  hypothesis is motivational rather than cognitive. We propose that the consequences
  of each bet include, besides the monetary payoffs, the credit or blame associated
  with the outcome. Psychic payoffs of satisfaction or embarrassment can result from
  self-evaluation from an evaluation by others. In either case, the credit and the blame
  associated with an outcome depend, we suggest, on the attributions for success and
  failure. In the domain of chance, both success and failure are attributed primarily to luck.
  The situation is different when a person bets on his or her judgment. If the decision
  maker has limited understanding of the problem at hand, failure will be attributed to
  ignorance, whereas success is likely to be attributed to chance. In contrast, if the decision
  maker is an "expert," success is attributable to knowledge, whereas failure can sometimes
  be attributed to chance.


So basically by "motivational explanation" they refer to is the posited existence of a "psychic payoff of satisfaction or embarrassment" in addition to the concrete/monetary one. In my view, the existence of a "psychic payoff of satisfaction or embarrassment" could also be a called a "[more] emotionally charged decision".
There's been rivers of papers on the role of emotion in decision making, e.g. see Lerner et al. (2015) for a review.

By the way (and as an aside), there's a newer paper in this area by Klein et al. (2010) saying that Heath &amp; Tversky basically confounded two effects:


  According to the competence hypothesis (Heath &amp; Tversky, 1991), this pattern derives from favorable perceptions of one’s competence. In most past tests of the competence hypothesis, ambiguity is confounded with personal controllability and the source of the ambiguity (e.g., chance vs. missing information). We unconfound these factors in three experiments and find strong evidence for independent effects of both ambiguity aversion and competence.


Klein et al. seem very careful not to call controllability anything else... or advance any hypotheses for what's behind it, but there's one older paper by Harris (1996) saying


  Evidence that perceived controllability is associated with optimistic bias is reviewed. There is sufficient evidence that people are optimistically biased about
  negative events they perceive to be controllable. More work is needed to establish
  whether this is also true of positive events. There is evidence also that people often
  hold beliefs of their personal superiority on control-related dimensions. More
  experimental studies testing hypotheses about the mechanisms linking perceived
  control and optimistic bias are required. Future work should also expand the
  sophistication with which control is treated conceptually and empirically and the range of potential mechanisms examined


Wikipedia's article on illusory superiority says that "Illusory superiority's relationship with self-esteem is uncertain" (and does cite some studies in support of this statement).

It's also interesting that perceived controllability affects quite a few other things, including neural pain response; see Salomons (2004).



And back to the general question: affect and emotion are not exactly the same thing:


  Core Affect. This term is defined as a "neurophysiological
  state consciously accessible as a
  simple primitive non-reflective feeling most evident
  in mood and emotion but always available to
  consciousness" (Russell &amp; Feldman Barrett, 2009,
  p. 104). Examples of core affect include pleasure
  and displeasure, tension and relaxation, energy
  and tiredness. A person experiences core affect constantly,
  although the nature and intensity of affect
  vary over time. Core affect can be a component
  of emotions and moods (defined next), but it can
  also occur in pure, or isolate, form. For example,
  according to Russell (2003), "pride can be thought
  of as feeling good about oneself. The 'feeling good'
  is core affect and the 'about oneself' is an additional
  (cognitive) component" (p. 148). As explained next,
  this qualifies pride as an emotion. 


Since what Heath &amp; Tversky talk about (motivation-wise) is basically an increase/decrease in feeling good about oneself (via satisfaction or embarrassment), technically I could have said "[core-]affectively charged decision" (instead of "emotionally charged") but that sounds really like a mouthful.

Also, even the broader class of affective phenomena has been disputed as being distinguishable from cognition, although this I think is a minority view. For a broader discussion of alternative viewpoints see "On the interdependence of cognition and emotion".


  We open the article by discussing three classic views for the independence of affect. These are (i) the affective independence hypothesis, that emotion is processed independently from cognition, (ii) the affective primacy hypothesis, that evaluative processing precedes semantic processing, and (iii) the affective automaticity hypothesis, that affectively potent stimuli commandeer attention and evaluation is automatic. We argue that affect is not independent from cognition, that affect is not primary to cognition, nor is affect automatically elicited. The second half of the paper discusses several instances of how affect influences cognition. 


See also "Mechanisms of motivation–cognition interaction", which details the way motivation has been studied in psychology, which is to say in a somewhat fragmentary way:


  The preceding sections highlighted the differences in how motivation is defined and investigated in various subfields. In animal behavioral neuroscience, the emphasis is on learning and conditioning processes, using primary incentives (food, liquid, and sexual stimuli) and measuring simple behaviors (physiological reflexes, response rates, and stimulus preferences). In social and personality psychology, the emphasis is on the pursuit of temporally extended goals involving high-level incentives (power, achievement, and affiliation) and assessing self-reported beliefs and goal striving behaviors. In cognitive neuroscience and adolescent developmental research, the emphasis is on neural representations of incentive value, typically using monetary rewards, and assessing how these modulate effortful cognitive processing. Finally, in cognitive-aging research, there is an emphasis on emotion–cognition interactions, using affectively valenced stimuli and measuring attentional and memory biases.


Unlike the cognitive/affective case, I have not seen a detailed discussion where motivation and cognition are contrasted... except this 1980s article: "Attribution bias: On the inconclusiveness of the cognition-motivation debate". Basically, there the cognitive perspective is taken to mean information processing on the problem alone, whereas the motivational perspective allows for other variables such as self-esteem. I think it's in this paradigm that Heath and Tversky phrased their contrast of cognition vs motivation.