Isotrivial but non-trivial family of elliptic curves

One idea is to show that the total space of the family is rational. In other words, we want to show that the surface $$ \operatorname{Spec} \frac{k[x,y,t]}{(x^3-y^2-t)} $$ is rational. This can easily be seen algebraically. At the level of rings, $$ \frac{k[x,y,t]}{(x^3-y^2-t)} \cong k[x,y] $$ where we used the fact that $R[t]/(t-a) = R$ for $a\in R$. In this case, $R=k[x,y]$. Consequently, our surface is isomorphic to $\operatorname{Spec} k[x,y]=\mathbb{A}^2$, so it is rational. (If you prefer a geometric argument, you can project away from a point.)

This proves that the given family is not trivial: Indeed, the total space of the trivial family is isomorphic to the surface $\mathbb{C}^* \times E$ where $E$ is the elliptic curve $y^2=x^3-1$ (since all the fibers are isomorphic elliptic curves). This surface is not rational, because a rational map $\mathbb{P}^2\dashrightarrow \mathbb{C}^*\times E$ composed with the projection $\mathbb{C}^*\times E\to E$ would give a dominant map $\mathbb{P}^2\dashrightarrow E$ which would mean that $E$ is unirational. Any unirational curve is rational (by Luroth's theorem), which is a contradiction (because an elliptic curve have genus 1, so it is not rational).

Whenever you have a family of (generically smooth, but possibly singular) elliptic curves over a smooth 1-dimensional base $B$, the local monodromy of the family around a point $b\in B$ is completely determined by the fiber at that point. Roughly speaking, this monodromy tells you how the homology groups of the fibers changes as you move along a loop in the base $B$. In this case, since the fibers are elliptic curves, their first homology is $\mathbb{Z}^2$, and thus the monodromy associated to a loop in $B$ will be a matrix in $GL_2(\mathbb{Z})$. The point here is that if the family is trivial, then the monodromy will be trivial (ie "the identity" $[[1,0],[0,1]]$).

When $B$ is a curve over $\mathbb{C}$ (or a number field), the relationship between monodromy and the singular fiber is completely classified by Kodaira in one of his papers "On the Structure of Compact Complex Analytic Surfaces" (he wrote a number of papers with this title). (I believe the case of elliptic surfaces over rings of integers in number fields was done by Neron/Tate)

Thankfully, one of the key results can be found here:

https://en.wikipedia.org/wiki/Elliptic_surface#Monodromy

In your case, your singular fiber above $t = 0$ is a cuspidal cubic (Kodaira type II), which according to the table has monodromy matrix $[[1,1],[-1,0]]$. Since this monodromy matrix isn't the identity, you can deduce that your family is not a trivial family. (Note that the only way to get trivial monodromy is for the "singular fiber" to actually be smooth, ie Kodaira type $I_0$).

As Ariyan commented, it can be done by hand, but it's not ugly at all. Let $k = \mathbb C$. Any change of variables preserving the short Weierstrass form $y^2 = x^3+Ax+B$ is $(x,y) = (u^2x', u^3y')$ for $u \neq 0$ (cf. Silverman III.1). (This needs that the characteristic is $\neq 2,3$). The coefficients transform as $B = u^6 B'$. Thus, an isomorphism between $\{y^2=x^3-t\}$ and $\{y'^2=x'^3-1\}$ over $k(t)$ would give $u \in k(t)^\times$ with $u^6=t$.

So we really need to make the étale base change $k(t) \to k(t^{1/6})$ to trivialise the family.