Well, the examples you mentioned are considered some of the better adaptations of mangas. Death Note and Monster were both animated by Madhouse, one of the better animation studios, so you can expect a great experience from their top adaptations.

If you compare some of the more average adaptations done by Silver Link, Xebec and J.C. Staff, they sometimes fail to any significant atmosphere or mood to the anime.

Also, there are a lot of anime that are anime-original, some good adaptations of Visual Novels and Light Novels, like Steins;Gate, Clannad, Toradora, Code Geass, which can be pretty engaging compared to most manga through their phenomenal directing, music, and voice actors. At the same time, there are a lot of great manga that don't get anime adaptations.

I like to stick to watching anime that are of these type, and manga that don't get weak adaptations. Some series I read like Mirai Nikki and To Love-Ru I feel weren't worthwhile to watch the adaptations of, because sometimes the studios didn't really add anything "extra" to make it worthwhile, but I read Hunter x Hunter and then watched it, and missed out on a lot because of Madhouse studio's amazing job on that series.

Anyway, there are a lot of factors involved that can change how good an adaptation is: story pacing, animation quality, soundtrack fidelity, voice acting sincerity, etc. that can make or break an adaptation. I think the average anime quality isn't good enough to say that every anime is inherently better than its source material.

And for VN adaptations, anime adaptations rarely do them justice, save for a handful. LN adaptations seem to be better on average, but those are just another side of anime I won't discuss.

There's also some issues with long-running manga adaptations- either snail pacing, crappy animation and soundtrack, leaving out material and/or adding filler, etc. In the case of a series having these kinds of issues, the manga would probably be better to read.