Are there search problems that cannot be written as decision problems?
In terms of decidability, they are equivalent. Indeed, in order to turn a decision problem into a search problem we do the following:

Run the decision algorithm
If it outputted &quot;no&quot;, return &quot;no solution exists&quot;
Else, create a list called $Active$, and a counter $w\in\Sigma^*$.
Until you find a solution, repeat the following 5-6 steps:
Create a new instance of an algorithm that checks whether $w$ is a valid solution, and add this instance to $Active$.
For every instance in $Active$, run the algorithm one more step. If any one of them returned &quot;true&quot;, return the $w$ associated with it. Otherwise, increase the counter $w$ by one.

Intuitively, the word $w$ enumerates over all $\Sigma^*$, and for each one we check wether it is a valid solution. Since the decision algorithm must have returned &quot;true&quot;, there is a valid solution, say $w^*$. Since $w$ enumerates over all $\Sigma^*$, then it will also go through $w^*$ and thus will find the valid solution.

In terms of computation time, the two might differ. It depends on the language itself, and on the granularity of the time complexity you are trying to achieve. For example, $SAT$ has a known search to decision reduction that runs in polynomial time. This means that you can solve the search problem by a polynomial factor of the decision problem. But it wont help you if you are looking for a sub-polynomial factor instead.