Difference of 'dynamic panel (Nickell) bias' and the 'incidental parameter problem' in panel data? To me both of them are talking about how the estimation of parameters in panel data can be biased, but it is very unclear to me how are they related?
 A: They deal with estimating different parameters but indeed share common features:
Nickell (Econometrica 1981) bias:
The time demeaning operation of fixed effects in a dynamic panel data model
$$
y_{it}=\alpha_i+\beta y_{it-1}+\epsilon_{it}
$$
leads to a transformed regression model
$$
y_{it}-y_{i\cdot}=\beta (y_{it-1}-y_{i\cdot-1})+(\epsilon_{it}-\epsilon_{i\cdot})
$$
where dots indicate time averages. Here, error terms $(\epsilon_{it}-\epsilon_{i\cdot})$ and regressors $(y_{it-1}-y_{i\cdot-1})$ are correlated even as $N\to\infty$, where $N$ is the number of units in the panel. This can be shown formally, but essentially follows from the observation that $y_{i\cdot}$ contains future $y_{it}$ which are generated by past $y_{it}$ which, in turn, are generated by past $\epsilon_{it}$ which are contained in $\epsilon_{i\cdot})$ 
Hence, even as $N\to\infty$, the FE estimator will not consistently estimate $\beta$.
Incidental parameter problem:
The classical Neyman and Scott (Econometrica 1948) case is an an example that MLEs need not be consistent. Consider a random sample of size $N\equiv nr$, $$X_{11},\ldots,X_{1r},X_{21},\ldots,X_{2r},\ldots,X_{nr},$$ where we have $n$ subsamples of size $r$, $X_{\alpha 1},\ldots,X_{\alpha r}$, $\alpha=1,\ldots,n$ which are distributed as $N(\theta_\alpha,\sigma^2)$. Hence, each subsample has a different mean $\theta_\alpha$, but a common variance $\sigma^2$.
It can be shown that the MLE for $\sigma^2$ is given by
$$
\hat{\sigma}^2=\frac{1}{rn}\sum_{\alpha=1}^n\sum_{j=1}^r(X_{\alpha j}-X_{\alpha \cdot})^2
$$
One may show that
$$\hat{\sigma}^2\to_pE(S_\alpha^2)=\frac{r-1}{r}\sigma^2\neq\sigma^2$$
Hence, the MLE is not consistent as $n\to\infty$.
So they are related through the fact that both FE and $\hat{\sigma}^2$ inconsistent estimators, that however both are consistent as the "other" dimension also goes to infinity - $r$ in the incidental parameter problem and $T$, the number of time series observations per panel unit in the Nickell bias case.
Nickell shows the inconsistency to be approximately equal to 
$$
-\frac{1+\beta}{T-1}
$$
for $T$ "reasonably" large.
