Tarvos' biggest weakness is that she is ultimately predictable. The fact that she's so easily set off means that she's easy to pick on. When she gets angry she tends to get flustered, and sometimes her quick thinking backfires and leads her into trouble. She also has difficulty dealing with extreme emotions, and when confronted with something she doesn't understand, she'll ignore it and possibly exacerbate the situation, get frustrated and react violently, or possibly become extremely uncomfortable and withdraw. She tries to present herself as unshakeable, but that's not quite true, and when she gets shaken it tends to have long-lasting effects.
More curiously is that she dislikes any sort of physical contact. A pat on the back or a hand on the shoulder freaks her out and causes her to react violently (usually to push off the offending hand), but anything more has the opposite effect. Something as simple as a hug causes her to have what could be called a mental breakdown, where she is only concerned with fleeing. Anyone who gets in her way is merely shoved aside rather than greviously harmed. Depending on the type of contact she may disappear for days on end, not going home but not going anywhere that she can be easily found. Once she's recovered she may extract vengeance if she is up to going near the person. She has an instinctual fear of anyone who has done this to her, and will keep her distance from them until she no longer feels threatened by them. The idea of being in a physical relationship terrifies her, and she is incapable of understanding why anyone would do such a thing.
Although she represses her emotions to an extreme degree, every now and then they bubble up in one form or another. Sometimes she can deal with them, but other times they freak her out as she doesn't know what to do with them. While she considers herself a finely-honed vengeance machine, she hasn't dealt with enough emotions outside of her pre-programmed job and thus anything not relating to that is mostly unknown territory. Her reactions to these feelings will vary depending on what they are, but either extreme violence or extreme meloncholy are the most likely possibilities.
She is also extremely protective of her family, even though she's one of the youngest, and considers their vendettas her own when she finds out about them. If she finds out anyone has tried to harm family members, she'll go after the one who did it to indulge in her primary function a little. She firmly believes that one one is allowed to harm her family apart from her, and takes these things VERY SERIOUSLY when they come up, even if it's likely she'll get wiped out in the resulting conflict.
Due to being split into the image of her host, Tarvos actually has terrible eyesight and needs her glasses in order to see. These glasses are not replaceable as they are actually a part of her. Other glasses can help correct the problem, but will not completely restore her sight. If they're destroyed she'll take damage for it - black eyes and cuts around her eyes as well as terrible headaches - but the glasses will regenerate after a few days. I'm sure that a fun game was stealing her glasses and watching her wreak havoc in the wrong direction or something because the other Avatars are all bastards.
Oddly enough, Tarvos craves sour candy, especially lemon drops. She considers sour candy one of the few good things humanity has ever done, and can be found stealing candies from stores every week.