Dilaton is the generic name of the Goldstone Boson (GB) associated with spontaneous breaking of scale invariance. Any model that break scale invariance spontaneously will give rise to a dilaton. N=4 SYM for example has a moduli space and any modulus away form the origin will break conformal invariance, and a massless dilaton would appear. The Radion is just the 5D holographic realization of a 4D CFT that break conformal symmetry spontaneously.
As for any GB, the dilaton $\sigma(x)$ provides a non- linear realization of scale invariance $x\rightarrow x^\prime=x/\lambda$, $$ \sigma(x)\rightarrow \sigma^\prime(x)= \sigma(\lambda x)+\log\lambda $$ where the effective lagrangian reads
$$ L=\frac{f^2}{2}(\partial\chi)^2- k f^4\chi^4+\frac{(\partial\chi)^4}{\chi^4}+\ldots $$ with $f$ is the analog of the pion decay constant and $\chi$ is just shorthand for $\chi=e^{\sigma}$. Since $\chi$ transforms as a field with dimension $\Delta=1$, $$ \chi(x)\rightarrow \lambda\, \chi(\lambda x) $$ the action $\int d^4 x L$ is left invariant under the scale transformation. In practice, any term with dimension 4 in the lagrangian must be written down. The only difference wrt to an ordinary GB is that the symmetry is a spacetime symmetry so that acts also on $x$. Analogous expressions hold for the full conformal group tang simply forbids certain operators that are allowed by scale invariance only. Notice that a pure quartic potential is allowed in principle (although it would require a curved space for consistently be non-vanishing\ldots)
Let's move now to the radion that arise in extra dimensional theory. To show that it is just a special realization of a dilaton from extra dimensions, we derive its effective action and matches to the one for the dilaton. Let's take the <PERSON> model that is a slice of 5D AdS space
$$ ds^2=\frac{L^2}{z^2}\left(dx^2_4-dz^2\right) $$ between an UV brane and an IR brane at $x=z_{UV}$ and $z_{IR}$ respectively. 364
$L$ is the AdS curvature. Of course, one of the isometries of AdS is $$ x\rightarrow x/\lambda\,,\qquad z\rightarrow z/\lambda $$ which is broken softly e.g. by the IR brane that gets moved $z_{IR}\rightarrow z_{IR}/\lambda$. There are various ways to get the effective action for the radion. One possibility would be looking at the metric perturbations and disentangle the spin-2 (graviton) from the spin-0 (radion) since both have zero modes. But, actually, a faster and more intuitive way comes from remembering that the radion is supposed to give the size of the extraD. In fact, it is clear that $z^{-1}{IR}$ that breaks the scale isometry has to be identified with the vev of $\chi$. We can thus integrating out the bulk of the extra dimension but letting the IR brane free to move and bend, so that you end with a 4D effective action for $z{IR}$ $$ L=-\frac{1}{M_^3}\int_{z_{IR}}^{z_{UV}} dz \sqrt{|g|} R+\mbox{boundary terms} $$ where the boundary terms contain the localized actions on the branes and the <PERSON>-hawking term. The resulting action is exactly the one for a dilaton up to identifying $\chi\sim z^{-1}{IR}$. For example, the bulk and the boundary terms give rise to an exact quartic potential (after using the bulk eom's that link the 5D cc $\Lambda _5$ to the 5D curvature $L$, and the 5D Planck mass $M$) $$ V=k z_{IR}^{-4}\qquad k\propto\Lambda_{5}L-V_{IR} $$ that is indeed the only non trivial potential that a scale invariant theory can admit. If the 4D space is actually flat then it results $k=0$ by consistently tuning the bulk cosmological constant $\Lambda_5$ and the localized IR potential to be the same. 649