There are no relevant differences between the manga and the first season of the Gunslinger Girl anime, as far as I am aware. However, the manga chapters are ordered different in comparison to the anime. The last anime episode of the first season matches the contents of chapter 8 of the manga, which is about half in the first omnibus-format book release, but following chapters do match contents of earlier anime episodes.
Since I haven't watched the second season of the Gunslinger Girl anime, I just looked up the ending of the anime and compared it with the manga. Since the anime ends in the first confrontation with Pinocchio, it matches chapter 15 of the manga, near the end of the first omnibus-format book release. I cannot say anything in regards to the other second season anime episodes.
My honest recommendation is to read the manga from the first chapter. The order might be a bit different, there might be a bit more background story, and certainly some manga scenes from these first chapters are referenced throughout the progression of the series. All by all, around ~16 chapters will only took you an hour or so, especially since you're familiar with the current setting.