The Big O

Both shows take place in a post apocalyptic setting featuring mecha that are of god-like proportions. Both of them also feature psychological aspects and religious symbolism(although it only has meanings in Big O and not Evangelion). 

Both series features science fiction themes with the involvement of mecha in combat warfare. Both series features a style of old school art that are nostalgic. There is also action, drama, and some psychological themes especially involving some of the later episodes and plot. Both series takes place in a post apocalyptic setting that also explores some religious themes.  

Post-apocalyptic settings with philosophical issues and giant robots. 

Both are mech shows that, while they certainly stay within some of the same conventions of said genre, have a uniqueness to them which make them feel very refreshing.  

Both shows have very similar mech designs, fight choreography, and characters. They also both have a similarly dreary tone and a psychological mind-bending plot. However, I feel that Eva gets a bit darker as it's story goes on, while Big O just gets more cryptic and enigmatic. 

Both these series are anime with a style that's more similar to American cartoons than anime itself. CB has its cowboy aesthetics, while Big O is often described as ''Batman turned anime''. They also share a strange, offbeat but at the same time melancholic atmosphere that's further enhanced by a jazzy soundtrack. If you liked one, then I strongly recommend giving the other a try.  

Both series contains a similar feeling with the main male protagonists being tough, independent, and collected. The main female protagonist from both series also has similar traits. Both series gives off that old school vibe with the jazz music and solid OST. Both series follows an episodic format with its themes. They both also have great dialogues, action, and drama, and science fiction themes. 

This one should be obvious. Not only did they air around the same time, but they also have the same studio and neo-noir feel (though "Big O" is more of a mech series than a western). The characters are also very similar: Spike===>Roger- Both are very laid-back, slothful, sarcastic people who often get annoyed by their allies but obviously care for them regardless. Faye===>Angel- Both have amnesia and are highly selfish, manipulative and seductive women who end up having a highly belligerent relationship with the main character. Jet===>Dastun- Very similar appearances, and a close but professional relationship with the main character that dates back several years. You could argue that Dastun is what Jet WAS like when he was still a cop. Imo, "Cowboy Bebop" is better than "The Big O" in pretty much every aspect, though there's still some enjoyment to be found in the latter. read more 

The Big O and Cowboy Bebop both have futuristic noir settings and have great blues and jazz music. Also, both shows have an underlying mysterious past that carries the mystique of the protagonists. 

Both series have mechs except that Big O is bigger. Both main characters have butlers and female assistants. Both have a noir atmosphere. 

Retro feeling, steampunkdieselpunk world, crime and detectives, giant robots battling. big o is better though, but steam detectives is also good for all steampunkdieselpunk lovers 

Both are very good steampunk shows that require at least one viewing. The robots both have a similar kind of design and mass as well. 

The Big O has so much references to Giant Robo, that you could think that they are both set in same universe. Actually, this is pretty good explanation for... oh, wait, no spoilers. But the atmosphere is completely different: Giant Robo has a classical "superheroic" plot, and The Big O is very close to noir drama. 

Both focus on overturning viewers' expectations and have great deal of philosophy themes. 

Both series are based in somewhat distopian worlds. Ergo Proxy is quite desolate world, where The Big O is just a little distopian (think Mad Max and Blade Runner in comprarison). Ergo proxy is more of a drama and psychological exploration, where The Big O is more stylised as a crime noir drama, even show design is based in 30´s america. Both are action based but with heavy dramatic scenes and quite significant character development. The Big O animation seems little old by todays standards, but given design of the show I think it is not that much of a hindrance. 

I feel like both are deep, dark, thought provoking shows. 

Both have giant robots and time loops. 

Both are primarily episodic neo-noir series that take place in a mysterious setting, and leave a lot of unanswered questions.