Let $M$ be a compact manifold and we are given two VFs $F,F' \in \mathcal{X}(M)$. We define a DS by $$\dot{x}=F(x)$$ and equivalently for $F'$. The $\omega$-limiting set is given by $$\omega(f,x)=\bigcap_{s \in \mathbb{R}} \overline{\{\varphi(x, t): t>s\}}$$
The two vector fields are topologically equivalent , denoted by $F \simeq F' \, $ if there exists a homeomorphism $h: U \rightarrow U$, mapping orbits of the first system onto orbits of the second system, i.e. $$\forall t \in \mathbb{R}, \ \forall x \in U: \quad \phi^{F}_t(x)=h^{-1} \circ \phi^{F'}_{\tau} \circ h(x),$$ with $ \tau: U \times \mathbb{R} \rightarrow \mathbb{R}, \quad \frac{\partial \tau(x,t)}{\partial t} >0 \quad \forall x \in U $. This means the time direction of the orbits is preserved.
My question is the following: Given that alle limiting sets are equal, $\forall x \in U: \ \omega(F,x)=\omega(F',x)$, can we conclude that $F$ and $F'$ are topologically equivalent?
No, you cannot conclude that $F$ and $F'$ are topoogically equivalent.
For example, on $T^2 = S^1 \times S^1$ there are many topologically inequivalent minimal flows --- minimality, here, means that the forward orbit of every point is dense, equivalently the only limiting set is the whole of $T^2$.
To see specific examples, represent the torus as $T^2 = \mathbb R^2 / \mathbb Z^2$. For each $r \in \mathbb R \cup \{\infty\}$, consider the unit speed flow on $\mathbb R^2$ whose orbits are lines of slope $r$. This flow on $\mathbb R^2$ is invariant under the action of $\mathbb Z^2$ and so descends to a flow on $T^2$ that I will denote $\varphi_r$. The flow $\varphi_r$ is minimal if and only if $r$ is irrational.
One can then prove (with some number theory and topology and continued fraction theory) that for each $r,s \in \mathbb R \cup \{\infty\}$ the flows $\varphi_r$ and $\varphi_s$ are topologically equivalent if and only if there exist integers $a,b,c,d \in \mathbb Z$ such that $ad-bc=1$ and such that $$r = \frac{as+b}{cs+d} $$ This defines an equivalence relation on $\mathbb R \cup \{\infty\}$ in which each class is countable (because the set of $4$-tuples $(a,b,c,d)$ is countable), but $\mathbb R$ is uncountable, so there are uncountably many equivalence classes. Hence there are uncountably many topological equivalence classes of minimal flows on $T^2$. To be very specific, $\phi_{\sqrt{3}}$ and $\phi_{\pi}$ are topologically inequivalent.