We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the order $\leq$.
My Question: Do we have any example of two compactification of $D(\mathfrak{c})$ that have two-point remainders but they are not homeomorphic.
Let $\kappa$ be any uncountable cardinal. Let $D_\kappa$ be the discrete space of power $\kappa$, let $p$ and $q$ be distinct points not in $D_\kappa$, and let $X=D_\kappa\cup\{p,q\}$. Let $P_0$ be a countably infinite subset of $D_\kappa$, and let $Q_0=D_\kappa\setminus P_0$. Finally, let $\{P_1,Q_1\}$ be a partition of $D_\kappa$ into two sets of power $\kappa$. For $k\in\{0,1\}$ define the topology $\tau_k$ on $X$ as follows:
$$\begin{align*} \tau_k=\wp(D_\kappa)&\cup\{U\subseteq X:p\in U\text{ and }P_k\setminus U\text{ is finite}\}\\ &\cup\{U\subseteq X:q\in U\text{ and }Q_k\setminus U\text{ is finite}\}\;. \end{align*}$$
Clearly $\langle X,\tau_k\rangle$ is a two-point compactification of $D_\kappa$. However, $\langle X,\tau_0\rangle$ is first countable at every point except $q$, while $\langle X,\tau_1\rangle$ is not first countable at either limit point, so $\langle X,\tau_0\rangle$ and $\langle X,\tau_1\rangle$ are not homeomorphic.