I have never read or otherwise studied the Principia; however, I think the general distinction to which Russell is alluding is still very much a recognized principle in modern (formalized) mathematics. Its basically the difference between a sentence $\varphi,$ versus the metasentence $\vdash \varphi$.
Conceptually, the distinction is best explained with reference to partially ordered sets (hereafter poset). In a poset, we can assert $x \leq y$ (intuitively, $x$ entails $y$). We may also have a meet-semilattice structure, in which case our assertions can be more sophisticated: we may write $x \wedge x' \leq y,$ intuitively asserting that $x$ and $x',$ taken together, entail $y$.
Note that $\wedge$ is a function, $\leq$ a relation.
Now furthermore, any given meet-semilattice may or may not admit the existence of a function $\rightarrow$ with the following property.
- $x \wedge x' \leq y$ iff $x \leq x' \rightarrow y$.
If such a function exists, it is unique, by this result. (If it is not clear what the above definition has to do with Galois connections, please comment and I will clarify.)
Anyway, if there is such a function (which I will call "implication"), then it can be added to the language (alongside $\wedge$ and $\leq$) to get a more expressive language. And we can prove the basic facts we expect from implication, such as modus ponens:
$$(x \rightarrow y) \wedge x \leq y$$
By the way, I recommend saying $\leq$ as "entails", and $\rightarrow$ as "implies", although this is not standardized.
Anyway, the point is that $\rightarrow$ can be conceived as an internalization of $\leq.$ Note that $\rightarrow$ is a function, while $\leq$ is a relation. Thus, spiritually, we can think of $x \rightarrow y$ as a statement internal to the language, while a formula like $x \leq y$ can (kind of) be viewed as part of the metalanguage. I am speaking very informally, here, of course.
Now I haven't really explained how $\rightarrow$ is an internalization of $\leq$, so lets do that. It turns out that if a function $\rightarrow$ with the property of interest exists, then so too does a top element, so long as we're working in a non-empty poset. (Hint: consider the expression $x \rightarrow x$). Denote the top element $\top;$ we can think of this as denoting unadulterated truthood. Furthermore, it can be shown that $x \leq y$ is equivalent to $\top \leq (x \rightarrow y)$. This is the sense in which $\rightarrow$ is an internalization of $\leq$.
Finally, lets switch to more logical notation. Instead of $\leq$, write $\vdash$ (this can also be articulated: "entails"). And lets move to greek letters, which can be thought of as denoting logical formulae. Furthermore, as shorthand for $\top \vdash \varphi$, let us write $\vdash \varphi.$
Then there is a clear difference between $\varphi$, and $\vdash \varphi$.
However, oftentimes $\varphi$ can be used as shorthand for $\vdash \varphi$, if the meaning is clear in context. Similarly, sometimes $\varphi \rightarrow \psi$ can be used as shorthand for $\vdash \varphi \rightarrow \psi$, or in other words $\varphi \vdash \psi$.
I think this is (at least conceptually) the distinction to which Russell is alluding.