In some contexts, both phrasal verbs have more or less the same meaning. When you “fight against” or “fight with” someone, you do battle against the individual.
*I refuse to fight against someone who is twenty years younger than me.
In the examples given, one could easily replace one phrasal verb with the other. It is also possible to “fight against” something as well. For example, one can fight against a disease, or one can fight against poverty. In these two cases, you are doing battle against a disease and poverty. “Fight with” has an additional meaning. When you fight with something, you use a weapon as an instrument in your fight. For example, in the old days people fought with spears and swords.
Nowadays people fight with guns. Also, when you “fight with” someone, it could mean that you are joining hands with the individual to fight someone or something. In other words, you and the other individual are joining forces and fighting a common enemy. “Fight against” does not have this meaning.
*I fought with him in Kashmir against the terrorists.