From what I've read in the introduction,
Fighting for the Confederacy contains personal experiences during the war and open critique since he wrote that only for his family and friends. Alexander then took
Fighting for the Confederacy as a base for
Military Memoirs, but he edited out everything personal and harsh and then added information about battles and campaigns he didn't take part in.
Military Memoirs was published in 1907, during his lifetime.
I've read
Fighting for the Confederacy as far as May/June 1863 and liked his descriptions and assessments of events, both negative and positive. If he critiqued someone, he didn't spare them. His personal experiences are sometimes amusing, sometimes shocking. I haven't read
Military Memoirs yet but intend to, but from the above I would hazzard a guess that there are some similarities but also many differences. According to the introduction for
Fighting for the Confederacy,
President Theodore Roosevelt acquired one of the first copies and hastened to write Alexander that "I have so thoro[ugh]ly enjoyed your 'Military Memoirs' that I must write to tell you so."
As for which is the better, it probably depends on what you like to read - more personal accounts or a more general view.