Are there circumstances in which BIC is useful and AIC is not? In the Wikipedia entry for Akaike information criterion, we read under Comparison with BIC (Bayesian information criterion) that

...AIC/AICc has theoretical advantages over BIC...AIC/AICc is derived from principles of information; BIC is not...BIC has a prior of 1/R (where R is the number of candidate models), which is "not sensible"...AICc tends to have practical/performance advantages over BIC...AIC is asymptotically optimal...BIC is not asymptotically optimal...the rate at which AIC converges to the optimum is...the best possible.

In the AIC talk section, there are numerous comments about the biased presentation of the comparison with BIC section. One frustrated contributor protested that the whole article "reads like a commercial for cigarettes."
In other sources, for example in this thesis appendix, the tenor of the claims for AIC seem more realistic. Thus, as a service to the community, we ask:
Q: Are there circumstances in which BIC is useful and AIC is not?
 A: According to Wikipedia, the AIC can be written as follows:
$$
2k - 2 \ln(\mathcal L)
$$
As the BIC allows a large penalization for complex models there are situations in which the AIC will hint that you should select a model that is too complex, while the BIC is still useful. The BIC can be written as follows:
$$
-2  \ln(\mathcal L) + k \ln(n)
$$
So the difference is that the BIC penalizes for the size of the sample. If you do not want to penalize for the sample there 
A quick explanation by Rob Hyndman can be found here: Is there any reason to prefer the AIC or BIC over the other?  He writes:

  
*
  
*AIC is best for prediction as it is asymptotically equivalent to cross-validation.  
  
*BIC is best for explanation as it allows consistent estimation of the underlying data generating process.**
  

Edit: 
One example can be found in Time Series analysis. In VAR models the AIC (as well as its corrected version the AICc) often take to many lags. Therefore one should primarily look at the BIC when choosing the number of lags of a VAR Modell. For further information you can read chapter 9.2 from Forecasting- Principles and Practice by Rob J. Hyndman and George Athana­sopou­los.
A: It is not meaningful to ask the question whether AIC is better than BIC. Even though these two different model selection criteria look superficially similar they were each designed to solve fundamentally different problems. So you should choose the model selection criterion which is appropriate for the problem you have.
AIC is a formula estimates the expected value of twice the  negative log likelihood of test data using a correctly specified probability model whose parameters were obtained by fitting the model to training data. That is, AIC estimates expected cross validation error using a negative log likelihood error. That is,
$AIC \approx E\{-2 \log \prod_{i=1}^n p(x_i | \hat{\theta}_n)\}$
Where $x_1, \ldots, x_n$ are test data, $\hat{\theta}_n$ is estimated using training data, and $E\{ \}$ denotes the expectation operator with respect to the iid data generating process which generated both the training and test data.
BIC on the other hand is not designed to estimate cross validation error. BIC estimates twice the negative logarithm of the likelihood of the observed data given the model. This likelihood is also called the marginal likelihood it is computed by integrating the likelihood function weighted by a parameter prior $p(\theta)$ over the parameter space.
That is, 
$ BIC \approx -2 \log  \int  [\prod_{i=1}^n p( x_i | \theta) ] p(\theta)d\theta$.
A: Q: Are there circumstances in which BIC is useful and AIC is not?A: Yes. BIC and AIC have fundamentally different goals. BIC estimates the probability that a model minimizes the loss function (specifically, the Kullback-Leibler divergence); a BIC difference of .1 between A and B implies that model A is roughly 10% more likely to be the best model -- assuming you start with close to no information and have a large sample size. AIC, by contrast, measures how good a model is at making predictions -- a difference of .1 means (very roughly) that model A will be about 10% better at making new predictions than model B.
This means that BIC can be better if you want to know the probability that a model is true. AIC can't give you that; if you try using AIC in this way, you get inconsistent estimates (i.e. AIC will not always select the true model).
On the other hand, AIC will be better at minimizing the expected loss.
AIC and BIC have two fundamentally different goals (BIC tries to maximize the chances of picking the best model, while AIC tries to maximize the expected quality of the model you select).
