Usually the original material is the deepest and best, but I’ve come across a number of properties where subsequent adaptations are more fleshed out and polished. The Basilisk anime series (from the manga series which was in turn adapted from the genre-defining Kouga Ninpou Chou 50s novel) is one of them; the Mirai Nikki anime is another.
The major adjustment is Yuno’s portrayal – the manga lacks the crazy-Yuno horror shots that proliferate in the anime, and depicts a lot of her internal dialogue. Even her character design is cuter and less threatening. The bug-eyed perspective shots in the anime remind you of her instability and underline the ambiguity of her relationship with Yukito – that makes for a better, tenser thriller. The First is even able to tell the Fourth about Yuno in the manga (he doesn’t get the chance in the anime).
Almost as significant is how much the anime leaves unsaid. The manga is overly clear about the characters’ relationships: Akise’s intentions, the Tenth’s feelings about his family, Yuno’s insecurities with Hinata. The anime keeps the 3 diary holders’ approach more ambiguous, and only touches upon Akise’s investigation into Yuno, downplaying discussion of her real identity (the manga is clear about her impostor status). Most importantly, the manga gives away the big twist by explaining the consequences of the time leap.
The anime fleshes out the search for Yukito and Yuno with traps, uses more plausible gas than live burial, and makes the initial phone-grab much smoother. Yuno never delivers food in the manga, and in the anime, Yukito’s mother has an actual reaction to the police involvement, even though it’s pretty laissez-faire. Yukito’s chase of his father is shown, and the extra manga stories are incorporated into the TV series.
On the other hand, the manga has a sequence when Yukito discusses abandoning Yuno in order to read into the future for the vault password, and shows Yukito burying his father. Additionally, the manga is unclearer about what Kousaka is doing when Akise is being dismantled, how Yuno forces Akise to change the future and alter his own prediction, why Akise lets the others die, how the Eleventh finds out about his death.
The ending receives divergent treatments: the manga goes for tension in the final confrontation, making it look like Yuno stabs Yukki, whereas the anime is romantic, showing Yuno’s diary, her thoughts about kissing Yukki and being happy (perhaps that counterbalances the desolate world it shows). And the finale is very different too – the manga explains its happy ending, while the anime only hints at what happened.
In any case, Mirai Nikki is one of my favourite series – I think the anime version is the best, and should be seen first, with the manga to clear up any niggling questions.