Z-Transform vs. Fourier Transform convergence
There are several signals for which the $\mathcal{Z}$-transform doesn't exist but the (discrete-time) Fourier transform does. One important example is the complex exponential $x[n]=e^{jn\omega_0}$ extending from $-\infty$ to $\infty$. As a consequence, also sinusoidal signals only have a Fourier transform but no $\mathcal{Z}$-transform, as already mentioned in GKH's answer.

Another important group of sequences with only a Fourier transform but no $\mathcal{Z}$-transform are impulse responses of ideal frequency-selective filters, such as ideal low pass, high pass, or band pass filters. The same holds for the impulse responses of the ideal Hilbert transformer and the ideal differentiator.

In general, two-sided signals with a constant envelope (such as the complex exponential and sinusoids), and two-sided signals that decay only as $1/n$ (such as impulse responses of ideal filters) don't have a $\mathcal{Z}$-transform, but they can have a Fourier transform. The corresponding expressions for the Fourier transform may contain Dirac impulses, or they are discontinuous.