It seems there is a distinction between the eventually-zero sequences $$c_{00} := \{(x_n)_{n\in \mathbb{N}} \mid x_n \in \mathbb{R};\ x_m = 0 \text{ for all $m$ greater than some $k$}\} $$ and the finite sequences $$\mathbb{R}^\infty := \{(x_n)_{n\in \mathbb{N}} \mid x_n \in \mathbb{R};\ x_m \neq 0 \text{ for finitely many } m\}.$$ For the sake of completeness, the square-summable sequences are: $$\ell^2 := \{(x_n)_{n\in \mathbb{N}} \mid x_n \in \mathbb{R};\, \sum_n |x_n|^2 < \infty\}.$$
Originally, I don't see why there is any distinction, based on the following argument: Any finite sequence will necessarily be zero after its final non-zero element, hence it is also eventually-zero. Conversely, any eventually-zero sequence will have a final non-zero element $x_k$, hence the number of non-zero elements is at most $k$, which has to be finite. Every finite sequence is obviously square-summable, so this would then imply that $$c_{00} = \mathbb{R}^\infty \subsetneq \ell^2.$$
However, it then occurred to me that the assumption of a finite $k$ may not be true, based on the following construction: Given any integer $N$, define a tuple $x^{(N)} := (x_n)_{0 \leq n \leq 2N}$ with the following entries: $$x_n = \begin{cases}1 & \text{if } n \leq N\\ 0 & \text{if } n > N.\end{cases}$$ Then $\lim_{N\to\infty} x^{(N)} \in c_{00}$, but not in $\mathbb{R}^\infty$. This sequence is also not square-summable, which would suggest $$\mathbb{R}^\infty \subsetneq \ell^2, \quad \mathbb{R}^\infty \subsetneq c_{00}, \quad c_{00} \not\subseteq \ell^2 \text{ and } c_{00} \not\supseteq \ell^2.$$ So which is correct, or neither?
The way you've written the definitions of $c_{00}$ and $\mathbb{R}^{\infty}$, they are exactly the same set.
It's not super common - and so in most contexts should probably be explicitly stated - but in my experience usually the notation $\mathbb{R}^{\infty}$ is a specific topological space, i.e. this set is being equipped with a certain topology that comes from viewing it as the limit of the finite dimensional Eucldiean spaces (See Unit sphere in $\mathbb{R}^\infty$ is contractible? and https://mathoverflow.net/questions/119362/unit-sphere-in-r-infty-is-contractible )
Whereas $c_{00}$ might just be notation for a set and if you're doing analysis/topology then talking about the subspace $c_{00} \subset l^2$ is different from talking say about the subspace $c_{00} \subset l^{\infty}$.