What does "detect different things" mean? I'll take it to mean "have different Adams-Novikov spectral sequences".

For Landweber exact theories $E$, the situation is pretty straight-forward (and not very exciting): the ANSS associated to $E$ depends only on the height of the theory (more precisely, on the heights at all primes).

For instance, suppose $E$ is Landweber exact, elliptic, and that $\pi_*E$ is $p$-local. Then $p$ is the only relevant prime, and the only possible heights are $0,1,2$, as these are the only possibly heights for the formal group of an ellitpic curve. If two such $E$ and $E'$ have the same height, then the associated ANSSes are the *identical*. The spectral sequences compute the homotopy groups of the spectrum $L_hS$, ($h=$ the height). There's a map $S\to L_hS$ from the sphere spectrum to this, so I guess we can say that *$E$ detects the image of $\pi_*S\to \pi_* L_hS$*.

Some details are in Hovey & Strickland, "Comodules and Landweber exact homology theories". (They really only show that the spectral sequences have the same $E_2$-term if the heights coincide.)

If you drop the condition that $E$ be Landweber exact, the precise nature of this will be a bit more complicated, but one still expects elliptic cohomology theories to "detect" only information up to height 2.

Here's an example to your explicit question. Let $C$ be any elliptic curve over the field $\mathbb{F}_p$. This will have a formal group of either height $h=1$ ("ordinary curve") or height $h=2$ ("supersingular curve"). For any such $C$, there is a ring spectrum $K_C$ with coefficient ring $\pi_* K_C=\mathbb{F}_p[u,u^{-1}]$, with $u\in \pi_2$, which is complex orientable, whose formal group is the formal group of $C$ ($K_C$ is the "2-periodic Morava $K$-theory" associated to the formal group).

Clearly, any such $K_C$ is an elliptic spectrum. The ANSS for $K_C$ captures height $h$ phenomena in the homotopy groups of spheres. Specifically, it computes the homotopy of the spectrum $L_{K(h)}S$, which is certainly different depending on whether $h$ is 1 or 2.

(If you want Landweber exact theories, you can use the theory $E_C$ associated to the "universal deformation" of the elliptic curve $C$, which will have coefficient ring $\pi_*E_C= \mathbb{Z}_p[[a]][u,u^{-1}]$ with $a\in\pi_0$ and $u\in \pi_2$.)