I watched the anime and couldn't find a explanation as to why Annie spared Armin's life?
Why would she?
Annie, at least in her mind, is not a villain. She may be an antagonist to the main characters in the Anime, but she does not think of herself as a villain. She does not think of herself as evil.
She would not benefit in the least from killing Armin. The only members of the expedition she killed were the ones that threatened her titan form. She trained alongside them, and even had formed bonds of friendship with some of them. Even the explorer corps members were at a discomfort dealing with her, after learning what she did.
Her main goal was to get Eren and deliver him to her masters/employers (it is still unclear who charged her with the kidnapping in the manga - as of chapter 66).
So, her reasons for not killing Armin may be:
- Armin was harmless. His vertical maneuvering skills were poor, and even if he were to attack her titan form, she would easily squat him.
- She does not think of herself as a monster, or even that she is evil, so she has zero motivation for random killing.
- More on her self-image, she is clearly a human that can change into titan (in contrast to Ymir who was a titan for 60 years before gaining the power of shifting). She has a good life with the MP Brigade. She is not a psychopath.
She may have some feelings of friendship or camaraderie towards Armin. Remember Armin has a very likeable personality. He gets along well with everyone.
Regarding her personality, Shingeki no Kyojin Wiki says:
Yet, she somehow does seem to hold a strange fascination and even feelings of respect towards people that have a deep sense of duty and righteousness
And it seems to me that Armin falls under that category.
Maybe killing Armin at that time would take more time than just confirming he is not Eren, and she is in a hurry to complete her goal.
- Armin was not alone. Killing him (remember that people actually LIKE Armin) could usher the other explorer corps around into a battle-frenzy to avenge his companion (maybe not likely, but who knows).
Reasons for killing Armin at that moment? Can't think of any.
She is not a ruthless killing machine. She has a goal in mind along with
Berthold and Reiner, which is not yet fully revealed.
SPOILERS IF YOU HAVE NOT READ THE MANGA:
Reiner was present with Armin. If she killed Armin she would have to kill Reiner as well because it would look extremely suspicious that Reiner got out of the fight alive while Armin died (since Reiner is physically superior and he would have to fight until the end if one of his companions is being attacked) BTW, Reiner is the Armored Titan.
Besides the above point: she does have compassion for her friends (as long as that does not conflict with her ultimate goal). When looking into Armin's scared face, she hesitated.
However, I do not think Annie spared his life because Armin was weak. She has killed a lot of other weak members. The only two reasons that I strongly believe she did not kill Armin are the above.
This topic seems to be deliberately not clarified by the manga writer and has been the subject of discussion on several forums for years. Based on that, the following is not an exact answer, but more like a theory about one of the reasons she didn't kill Armin other than friendship reasons and such.
I think it was related to her mission. As we know that several personnel had different information regarding Eren's position, and Annie and her teams already know that Armin is a smart and clever person based on their observations as well as his experiences on Battle of Trost, and might be able to help them locate Eren. Knowing Eren is not on the right wing as informed to Reiner, they improvise and ended up with Reiner indirectly making Armin to think about Eren's whereabouts