The Planetes (See also ΠΛΑΝΗΤΕΣ) manga is substantially different from the anime. Many of the themes are the same, but the plot and storylines are different; there are stories in the manga that do not appear in the anime, and vice versa. There is also less initial character development. The story jumps right into the mix in the manga.
Some other notable differences include Tanabe's absence from the beginning of the manga. There is also much less time devoted to setting up her relationship with Hachimaki. A few pages tell you that months go by, and rather suddenly you're expected to accept that the two of them are getting along much better now.
I also don't care for the character design in the manga too much.
The manga is quite a bit more serious, with much less comic relief, and there are fewer characters, so it focuses more on the main leads.
I am also displeased with the editing in the Tokyopop volumes. It occasionally contains stupid errors like mixing up "your" and "you're" and there are some typos. There are also dumb errors like lines about "finishing all 12 bottles" when the frame clearly shows cans. On the whole, it frankly makes it feel rather amateurish.
Overall, I do like the Planetes manga, although I like the anime a lot more, despite the fact that it can be quite silly on occasion. I suppose the manga is a worthwhile read if you've already watched the anime—some of the stories are quite different, so it stays fresh and interesting even if you've already watched the anime.
However, I would not recommend reading the manga first before watching the anime as the manga spoils nearly all of the best parts of the anime. Moreover, the storylines and events contained in the anime but absent from the manga are sometimes kinda annoying. I.e., if you read the manga first, you'll spoil the best parts of the anime and have a lot less good stuff the look forward to, but if you watch the anime first, you may spoil parts of the manga, but at least there are interesting different stories in the manga that will still be new.