What are the differences between the Akame Ga Kill! manga and anime? First of all, why was the anime completed before the manga? As far as I am aware, the anime is ahead of the latest manga chapters by a large margin.

I've heard that the anime actually differs from the manga; does this mean we might get a remake like Fullmetal Alchemist: Brotherhood for Fullmetal Alchemist?

Furthermore, will the manga continue on with Akame's story? Will it explain some of the plot holes that were left unanswered in the anime?
Difference between the anime and manga of Akame ga KILL!
I have listed out some of the differences between the anime and manga of Akame ga KILL! in this answer. Let me reproduce it here for posterity.

The story is generally the same up to episode 17/chapter 33. However, the anime begins to have minor deviations from the manga from episode 18/chapter 34. After chapter 38, which corresponds to episode 19, the anime totally skips more than 10 chapters starting from chapter 39, except for this plot point in chapter 43, where

 ... Susanoo died fighting Esdeath to let the rest of the Night Raid escape. Originally, this happened during the mission to assassinate Borick, which spans from chapter 40 to chapter 43. In the anime, Susanoo's death is moved to episode 21 during the rescue for Tatsumi instead, and Borick's assassination is trivialized at the end of episode 19.

Episode 20 restarts from second half of chapter 50, but doesn't follow the manga too closely, since the manga introduces many new characters during the Wild Hunt arc (from chapter 44 to 48), and several of them affect the story beyond Wild Hunt arc.
Most of the differences between manga and anime adaptation come from this 10-chapter gap.

Possibility of a remake ending
It's not unprecedented and even common for the anime to have an original ending, when the anime runs out of source material. Heck, for the case of Fullmetal Alchemist, the 2003 anime branched off from the manga half-way and followed its own original storyline.
For the case of Akame ga KILL!, it is extremely unlikely that it gets a remake. Unlike the case of Fullmetal Alchemist (2003) anime where the plot in the manga was still developing, Akame ga KILL! manga is currently on its way to the ending. The anime's ending also prohibits any logical extension to the story, and an anime continuation would require a completely new arc in the manga, plus great demand from fans.
Since the manga is currently on-going, only time will tell whether the manga has the same ending as the anime or not, or whether there is any extension to the story of the characters who survives.