You should add character shadows. That’s a big advantage of using a third person character model in first person view. (But in a first person only controller you can add a third person character model for shadows only, which is a very good option by the way because it doesn’t need to have lot of details or impressive textures or anything.)
You will need a character model, which could be a problem at least you already have one or can make your own. With a first person only controller you can avoid this, and it might be even more “immersive” because you can describe the character as you want or let the player do it for you. (It encourages the use of the imagination, which is no a small thing.)
Usually, character controls are clunky as hell, even in AAA big productions. But first person only has its problems too: walking is usually too fast and unrealistic, turning around doesn’t feel right, character lacks sounds like cloth, etc, but if you are creative and imaginative enough you can do it better.
I like both things, but usually dislike how are implemented most of the time. So it doesn’t really matter. I think you should think about what kind of experience you want to give to the player, what you want she or he feels when playing the game, how interact with the world, and use the one which gives you that better or, in case is important, would make things easy and/or faster for you, the creator. Once you know that both approaches are good, it really doesn’t matter as much.