My analysis
After seeing all of Cowboy Bebop and Trigun, as well as plenty of my friends, we have always had a large argument of some sort revolving around which was the better anime. I'm here to present both sides with both* pros and cons, although I do side with Bebop. I found it was far superior. (And hopefully provide some useful information for those of you who have not seen either.)
Trigun.
I saw Trigun before Cowboy Bebop, I think around last year sometime in October. This was my big step into anime, because all that I had watched regulary was DBZ and Ranma 1/2, and a few anime movies. It opened a gate into a new view, for me, of how talented anime artists are. The character design is amazing, and some are very complex. Parts of the animation can seem a little luke warm at times, but over all its very good compared to others i've seen. (such as DBZ with the repeatative scenes etc...none of those in Trigun)Having said that, How about the nity grity stuff? 
As for the storyline, I found it to be pretty well thought out, and simple enough for younger ages to follow. The time set is a futuristic 'Wild West' theme, with a $60,000,000,000 Wanted Man on the loose, completely annihilating entire cities in his path. The wanted man is? Vash the Stampede of course! Vash is the main character of Trigun, but does not, I repeat, *NOT* live up to his rough and reckless stero-typical personality. You'll find his only goal is LOVE AND PEACE!, (*giggles* Sorry, a little Trigun humor there.)
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Vash is very charming to viewers, but has a deep side to him concerning his brother Knives who is plotting the destruction of mankind.
Trigun is 26 eps long, and keeps you guessing until the very end. With a deep storyline that anyone* could get sunk in, next to Vash's stupidity/humorous personality, and AWEFUL luck, Trigun keeps a pleasent comedy relief throughout the series, and provides enjoyment for anyone who can take some of the language, gun usage, and a few of Legato's crude moments. ^.^
Trigun is rated 13 and up for just that reason. To find out more about Trigun, visit this link:
http://www.animetric.com/tuv/trigun.html
Cowboy Bebop.
Before I go any further I want all of you, yes, YOU, to open up Kazaa or w/e you use to download things, and download the song,'Tank' from this amazing anime series. WOW!!!!!!!! Just mentioning it gets the catchy tune stuck in my head. Cowboy Bebop has a number of awesome jazz songs and themes, especially the well-known title song Tank. ( ^.^ I also recommend 'Stray Dog Strut' and 'The Real Folk Blues')
Music, A+++++ For Bebop. 
Character Design. Spike Spiegal is just an average bounty hunter, right? Wrong. Well, I suppose you'd consider him average if you think swallowing cigeretts WHOLE is average. I'll get more into the storyline and things later, but it's more important to comment on the animation first. I have not seen a more brilliant animated anime. Hmm...animated anime that's kind of funny sounding...oh well don't listen to the monkeys in my head. 0.o Anyway! Bebop is the best animated show i've seen, even the spaceships are sweet. It's crisp and smooth, and very eye pleasing. The fight scenes are what make this show great, as well as the common chase scenes. I think most anyone would love this show at LEAST for that....
For the Storyline, it's set in a much more futuristic setting compared to Trigun, with multiple planets and plenty of space travel/space stations. The worlds are extremely futuristic as well, some resemble entire worlds with populations of Tokyo or New York. Of course the main goal in this anime is to uncover Spike's past, and somehow work it into his tight budget as he and Jet (his partner who flys with him in the Bebop, their ship) struggle to gain money for bell-peppers and beef! :P The storyline gets very deep, much deeper than Trigun's for me at least, and hits harder. Mainly because this anime seemed to have less humor, but only when Ed(another character) wasn't around.
Cowboy Bebop is rated 13 and up I think* but am not entirely sure, but one thing IS for sure, I would not recommend this anime to younger people, since it is aimed toward older folk, due to language and violence.(and wonderful boob shots from Fae. Thanks girl. *rolls eyes* O well whatcha gunna do.)But for anyone who can handle that, this anime is a must. I can almost garuentee you you're going to love it. no, I CAN.
As for which do I like better? Bebop was based for much older crowds, and seemed to handle more adult issues. But, they were both awesome, and both house two of my favorite villians, Legato (Trigun) and Vicious (Cowboy Bebop), I can't come to a conclusion on which is cooler. I suppose you're going to have to make that decision, because I do not think I ever will.
My analysis