there are some difference between this two character in usage:
打仗(√) 打战(×) at war, fight , fight a battle
战争(√) 仗争(×) war (battle, fight)
when the character 仗(noun) means 'war'(there'er other meanings of 仗) , normally as the word '打仗'(verb):
那时两国正在打仗 At the moment the two countries were at war. 打胜仗 win a battle (仗 can be a fight ,a battle) and it doesn't particularly indicate a location,time,or the scale of war.
战(noun & verb ) 战争(noun): war 战胜(verb) :defeat,win ,beat, overcome
when you indicate a certain war like WW II(二战) Civil War(南北战争),you must use the character 战.
and in Chinese, 战 usually be combined to the items which associated with war, like 战机(warplane) 战马(steed) 战场(battlefield) 海战(naval battle) 陆战(land battle)...
When you don't know which one to select, just take 战 :p