Often times a manga gets enough recognition that an anime adaptation comes to fruition. But there are people who always tend to prefer the reading the manga than the anime series, are you one of them?
Personally, I find a majority of anime series to be relatively lacking in comparison to their manga counterpart. This is probably because often times studios like to cut out certain parts completely or rewrite them in a way that's different from the manga series \: What's your opinion? Do you prefer seeing the action and story come together?
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Honestly, I think that in general, whichever one came first is the better one. In most cases the manga comes first, but I have seen exceptions. Blood+ and Samurai Champloo are two examples. The Samurai Champloo manga was meh, and the Blood+ manga just reads like a cheap knockoff of the anime.
In most cases though, the manga is the better one. Like you said, the anime usually cuts out a lot, and in my experience, they almost always cut out the best parts.
Generally I agree with what has been said so far; the original format (more often than not the manga) is often better than the adaptation. That being said however, for me there is also the factor of genre to be considered. In cases like Death Note for example, where most of the appeal lies in the dialogue, the manga is better because it gives you time to think; whereas more action oriented stories might "come alive" better in an anime.
A second factor than also might influence my preference would be art style. I love the Saiyuki manga and think Minekura's work is wonderful. When that is translated into anime, even though I love the voice acting, 9 times out of 10 the art suffers and is made really generic. In other words, the greater the attention to detail, the more likely that the manga will be superior.
Anyways, it pretty much works like this... if you want story, you go for manga, if you already know the story, anime is for seeing characters moving and voiced and for animation (I'm looking at you, SHAFT).
For me it's like this.
I watch anime and if I really like it, I continue with manga to continue with the story.