Confirmation bias is the tendency to search for, interpret, favour, and recall information in a way that confirms pre-existing beliefs or hypotheses (WikiPedia; Plous, 1993)
The affect heuristic is a mental shortcut that allows people to make decisions, and solve problems quickly and efficiently, in which current emotion - fear, pleasure, surprise, etc. - influences decisions.
In other words, it is a type of heuristic in which emotional response, or "affect" in psychological terms, plays a lead role. (WikiPedia)
Or in the words of Slovic,
As used here, ‘‘affect’’ means the specific quality of ‘‘goodness’’ or ‘‘badness’’ (Slovic, et al. 2007)
They can sound similar, and in a way they can be; but generally they are different. With confirmation bias, I would be looking for something to back up a belief I “know” is correct, while I may or may not like this belief,
it is the only correct answer!
This is different to the affect heuristic, where I don’t like the proposition put forward by others and so I am going to look for evidence to back the ideas I like.
--Edit for further clarification after comment--
You could say
You can search for evidence to confirm something is right if you believe it's right, and you can search for evidence to confirm something is wrong if you believe it's wrong
and that is true. That would be Confirmation Bias if you are searching purely for evidence to prove your belief based on what is known. Let's look at the difference in another way.
Scenario where there are two opposing points of view ($X$ and $Y$).
$X$ has a conclusion based on plenty of evidence which is corroborated by many unbiased studies.
$Y$ has a conclusion which is also based on plenty of studies which corroborates each other.
However, there were flaws in some studies and others were biased due to undeclared financial interests with some participants.
When there is no bias, you are completely open minded and you would be taking an unbiased viewpoint on the evidence presented, therefore it would be fair to assume that $X$ is true and $Y$ is false. When looking for more evidence, you are looking for evidence for and against both $X$ and $Y$.
When there is Pure Confirmation Bias, you totally accept $X$ being true, and it fits with your beliefs and way of life, and when looking for more evidence, you are looking to strengthen that truth with more evidence to back $X$ and more evidence to refute $Y$.
When there is Confirmation Bias with Affect Heuristics, this can be where you don't like viewpoint $X$ because it doesn't fit with your beliefs. You can accept $X$ being true, but there is the counteracting wish for $Y$ to be true.
There can be affect heuristics involved when looking for any strong counteracting evidence, yet you are still accepting that $X$ is true, so whilst looking for counteracting evidence, you are also looking for evidence confirming $X$. You could be forgiven for thinking that this is unbiased research, but it is not unbiased, because you wish for $Y$ to be true as you have a vested interest in $Y$ but you also accept $X$.
When there are pure Affect Heuristics, you cannot accept $X$ as true because of your beliefs, or because it adversely affects your way of life, and when looking for more evidence, you are looking for evidence to back $Y$ and plenty of evidence to refute $X$.
Plous, S. (1993). The Psychology of Judgment and Decision Making. New York: McGraw-Hill p.233
Slovic, P., Finucane, M. L., Peters, E., MacGregor, D. G. (2007). The affect heuristic. European Journal of Operational Research 177(3),1333–1352