When there is both an anime and a manga of the same story the natural question is which of the two should you experience before the other. The simple answer is usually to seek out the original first and then look at the other one if you have the time and money.
When a title is originally an anime seeking out the manga is usually only for those who are gluttons for punishment. They tend to be pale imitations of the original with truncated narratives and inferior storytelling. There are some notable exceptions but they are mostly the exception that proves the rule.
Titles that are manga first fare better when translated to anime. They tend to be hit or miss if they can live up to the original but good anime adaptations of manga are hardly shocking. The real rarity is the anime that surpasses its origin. If anything the anime are usually just very competent direct translations that add little to the story but also don’t lose much either. Overall your best bet is to stick to manga and maybe watch the anime if it has some cool fights or pretty scenery. It also means if you saw the anime there is little reason to read the manga.
This formula is a quick criterion that does a pretty good job of making effective use of your time. It is important to note that there are some special examples that stand out as prime examples of flying right in the face of that simple rule of thumb. The rare case where the anime and manga are both are each brilliant pieces of art in their own way and worth experiencing twice. Each version tells the same story but is able to do it in a way that is complementary to both versions. Akira and Nausicaä of the Valley of the Wind are the two titles that come to mind when talking about this. People often prefer one over the other but still recommend that you see both. I would say without hesitation In This Corner of the World is also on this short list of titles that are worth experiencing in both of its forms.
Suzu is an amateur artist and a space case who lives with her family in Hiroshima. When she is married off to a young man in Kure who works at the naval yards she moves in with his family there. The normal adjustments of moving in with your in-laws is exacerbated by the shadow of WWII in the background. Slowly Suzu’s somewhat idyllic rural life is transformed by the specter of conflict around her. As the bombings, shortages, and pressure of the war increase she valiantly tries to survive with a modicum of sanity and normality. This is the story of one ordinary woman surviving great hardships with grace and dignity.
Not factoring in anything else In This Corner of the World is a fascinating look at a civilian population of an Empire at war. Suzu and her family start living very simple lives but as the series goes on the war gets closer and closer to them and the machinery of empire takes more and more from them. The transition is slow but noticeable. At the same time, the story is not just a dark meditation on the horrors of war. Despite the sacrifice, propaganda, and terror there is still humor, warmth, and strength. The extremes of emotion in the manga make both halves much more meaningful.
I think the most interesting part of the story you only really get in the manga is the friendship between Suzu and Rin Shiraki. The anime of In This Corner of the World was a labor of love that was only completed thanks to the perseverance of the staff and crowdfunding from fans who believed in the project. While it was amazing that the anime was made it had to cut some story to fit within its budget. One of the major cuts was Suzu and Rin. While the movie has a major scene with the first meeting of Suzu and the young prostitute Rin mostly disappears from the story after that point. In the manga, Suzu and Rin meet several times and develop a complex friendship. I understand why it had to be dropped in the anime but in real adds a level of depth to Suzu only seen in the manga.
The manga also plays with art styles. Since Suzu is an artist the story is often told via her art and so her more flighty moments are wrapped up in her softer dreamy style where are darker turns in the story change the way the series is drawn. The art of the book is often a subtle barometer of Suzu’s emotional state and her attachment to reality. It is a clever conceit and a real charm of the book.
This is a powerful story really worth experiencing however you can. If you have never experienced the story in any way this is a complete and powerful story in a single book. If you have already seen the anime the is another dimension of the story to be experienced via the manga.