Examples where the uniqueness of the solution makes it easier to find The complexity class $\mathsf{UP}$ consists of those $\mathsf{NP}$-problems that can be decided by a polynomial time nondeterministic Turing machine which has at most one accepting computational path. That is, the solution, if any, is unique in this sense. 
It is thought highly unlikely that all $\mathsf{UP}$-problems are in $\mathsf{P}$, because by the Valiant-Vazirani Theorem this would imply the collapse $\mathsf{NP}=\mathsf{RP}$.

On the other hand, no $\mathsf{UP}$-problem is known to be $\mathsf{NP}$-complete, which suggests that the unique solution requirement still somehow makes them easier.


  I am looking for examples, where  the uniqueness assumption leads to a faster algorithm. 


For example, looking at graph problems, can a maximum clique in a graph be found faster (though possibly still in exponential time), if we know that the graph has a unique maximum clique? How about unique $k$-colorability, unique Hamiltonian path, unique minimum dominating set etc.? 

In general, we can define a      unique-solution version of any $\mathsf{NP}$-complete problem, scaling them down to $\mathsf{UP}$. Is it known for any of them that adding the uniqueness assumption leads to a faster algorithm? (Allowing that it still remains exponential.)
3-SAT may be one such problem.  Currently the best upper bound for Unique 3-SAT is exponentially faster than for general 3-SAT.  (The speedup is exponential, although the reduction in the exponent is tiny.)  The record-holder for the unique case is this paper by Timon Hertli.

Hertli's algorithm builds upon the important PPSZ algorithm of Paturi, Pudlák, Saks, and Zane for $k$-SAT, which I believe is still the fastest for $k \geq 5$ (see also this encyclopedia article).  The original analysis showed better bounds for Unique $k$-SAT than for general $k$-SAT when $k = 3, 4$; subsequently, however, Hertli showed in a different paper that you could get the same bounds for (a slightly tweaked) PPSZ algorithm, without assuming uniqueness.  So, maybe uniqueness helps, and it can definitely simplify the analysis of some algorithms, but our understanding of the role of uniqueness for $k$-SAT is still growing.

There is evidence that Unique $k$-SAT is not too much easier than general $k$-SAT. The Strong Exponential Time Hypothesis (SETH) asserts there is no $\delta &lt; 1$ such that $n$-variable $k$-SAT is solvable in $O^*(2^{\delta n})$ time for each constant $k \geq 3$. It was shown in a paper of Calabro, Impagliazzo, Kabanets, and Paturi that, if SETH holds, then the same statement is true for Unique $k$-SAT.  Also, if general $k$-SAT  requires exponential time, i.e. there is some $k \geq 3, \epsilon &gt; 0$ such that general $k$-SAT cannot be solved in time $O^*(2^{\epsilon n})$, then the same must be true for Unique 3-SAT.  See the paper for the most general statement.  

(Note: the $O^*$ notation suppresses polynomial factors in the input length.)