TITLE: Cases where the number field case and the function field (with positive characteristic) are different
QUESTION [30 upvotes]: In number theory there is often an analogue between statements which holds over a number field (that is, a finite field extension $K/\mathbb{Q}$) and function fields (that is, finite extensions of the form $K/\mathbb{F}_q(t)$ where $q = p^k$ for some prime $p$ and $k \geq 1$). One of the most famous examples of such an analogue is the Riemann hypothesis. Weil showed that the Riemann Hypothesis holds for the function field case. However, progress on the function field case has not shed much, if any, light on the corresponding case for number fields.
An example of where the analogy breaks down (conjecturally) is the Brauer-Manin obstruction. For hypersurfaces defined over $\mathbb{P}^n(K)$ for some number field $K$, it is expected that most hypersurfaces $X$ of degree $d \geq n+1$ will be of general type, and hence neither satisfy the Hasse principle nor have its failure to satisfy the Hasse principle accounted for by the Brauer-Manin obstruction. Harari and Voloch showed that for function fields of positive characteristic, this is not the case: indeed in this paper (http://www.ma.utexas.edu/users/voloch/Preprints/carpobs3.pdf) they showed that the Brauer-Manin obstruction is the only reason Hasse principle can fail in this case.
Another case where the analogy is not convincing is ranks of elliptic curves. It is known that elliptic curves over a function field of positive characteristic may have arbitrarily large rank. However, this question is not known even conjecturally in the number field case. Indeed it seems that many experts disagree on this question. At a recent summer school on counting arithmetic objects in Montreal, Bjorn Poonen gave a take on a heuristic suggesting that elliptic curves over number fields should have bounded rank. Andrew Granville, one of the organizers, agrees with this assertion. However, other experts present including Manjul Bhargava disagreed. 
My question is, what are some other situations where one expects genuinely different behavior between the function field setting and the number field setting?

REPLY [10 votes]: I answered this question once already, but another example where $\mathbf Z$ and $k[x]$ behave differently when $k$ is a finite field was brought to my attention recently by Jeff Lagarias and it deserves being mentioned here as a separate answer.
Theorem 1. The group ${\rm SL}_2(\mathbf Z)$ is finitely generated.
Theorem 2.  For each finite field $k$, the group ${\rm SL}_2(k[x])$ is not finitely generated.
Theorem 1 goes back to the 19th century, with ${\rm SL}_2(\mathbf Z)$ having  generators $S = (\begin{smallmatrix}-1&0\\ \, 0&1\end{smallmatrix})$ and $T = (\begin{smallmatrix}1&1\\0&1\end{smallmatrix})$. Theorem 2 is due to Nagao ("On GL(2,$K[X]$)," J. Inst. Polytech. Osaka City Univ.Ser. A10 (1959), 117-121).
The two theorems of course also hold using ${\rm GL}_2$ rather than ${\rm SL}_2$.
To appreciate the special nature of the $2 \times 2$ matrix setting, Nagao points out at the start of his paper that for $n \geq 3$ the groups ${\rm SL}_n(k[x])$ and ${\rm GL}_n(k[x])$ are finitely generated.
Since a finitely generated group has only finitely many subgroups of each index, ${\rm SL}_2(\mathbf Z)$ has countably many subgroups of finite index. In contrast to that, ${\rm SL}_2(k[x])$ for finite $k$ has uncountably many subgroups of finite index.
Generalizations of Theorem 2 are in A. W. Mason, "Serre's generalization of Nagao's theorem: an elementary approach," Trans. Amer. Math. Soc. 353 (2001), 749-767 and H. Behr, "Arithmetic groups over function fields. I. A complete characterization of finitely generated and finitely presented arithmetic subgroups of reductive algebraic groups," J. Reine Angew. Math. 495 (1998), 79-118.