gn.general topology - The difference between a sequential space and a space with countable tightness

Just a partial answer.

For $beta mathbb{N}$ (the set of all ultrafilters on $mathbb{N}$ with the Stone topology) it is not hard to see that a sequence converges iff it is eventually constant. Hence any subset of $beta mathbb{N}$ is sequentially open -- and of course, $beta mathbb{N}$ is not discrete, so it cannot be sequential. Similarly, for the ultrafilters on $mathbb{R}$.

If I recall correctly, this 'trivial sequential convergence' holds in all extremally disconnected spaces -- this should be an exercise in the book 'Rings of continuous functions' by Gillman and Jerison (there is also a PDF/TeX-file with all exercise solutions freely available on the web somewhere).

Also, I could be wrong, but I think $beta mathbb{N}$ is not countably tight since its remainder is not (since there exist weak P-points). Maybe somebody else can confirm or reject.