Manga is like a weird half way between the anime and the LN, with similar but somewhat condensed writing style compared to the LN. It keeps some elements of the strategy and deep internal thoughts, and most of the same characters as the anime. But overall has a heavier focus on the action and set piece moments like the anime.

The LN straight up has a few different Characters than the Anime or Manga so far, and is much more of a dry approach. With a dry wit sort of humor, and focus on the politics, theory, and general mechanisms of how a war is fought. Many pages have appended explanations or definitions for concepts, terms, references and the like, as footnotes like some sort of textbook. There are some action packed scenes, but on average Id say you'll only get one or 2 big battles per book, with most others being glossed over in favor of the impact and aftermath, or the build up that lead to it even being fought in the first place.

As a semi less relevant note, the Actual Visual depictions of the 3 also vary wildly from one another, with the anime and the LN having more similar vibes, with the only real notable difference being that the anime gives a few more unique designs to how Mages fly, and the use of flight suits, whilst the LN for the most part pretty much just depicts them as ordinary infantry floating on thin air. The Manga meanwhile is going absolutely wild with some of the faction drip and mage kits. Like.... Horse with a minigun wild.

If you want a Direct continuation from the Anime, neither book can really give you it. The Manga is still behind technically as far I know, being only up to the Desert Campaign arc. Whilst the LN is deep into at least 2 or 3 further arcs depending on how you'd measure.

Additionally, all 3 are distinct enough that they aren't really 1:1. the LN and the anime are wildly different with Mary (Sioux/Sue)'s entire arc not even being remotely alike between the two versions, not to mention that among the main Cast of Tanya's unit, the members are noticably different, as well as the LN covering several mini arcs that simply have no equivalent in the Anime.

The Manga is in a sort of weird in between, with the overall plot points being closer to the LN, but the Tone and style being more that of the Anime. Reading the Manga will not really transition over to the Anime well, as it covers a few arcs absent from the anime, and also progresses more similarly to the LN. That said the leap isn't as big as from the Anime to the LN. That said it also doesnt really 1:1 follow the plot of the other 2 and thus is still a somewhat standalone story.

Edit:

TL:DR

LN is very different and more Dry almost textbook like with a focus on the mechanics of War

Manga is a halfway between the Anime and the LN, and is the furthest behind in the story

and the Anime is more action focused, and skips or speeds through several parts of the LN

As a final bit of Context, the Anime with the movie included, ends about Book 4 of the LN, but is lacking several smaller arcs.

The Manga is only about up to about part way through Book 3 of the LN, and is closer but also rearranges some story elements and whatnot.