Pros
Note: This is from an Engineer's perspective. SpaceX has been in a state of growth since I first started at the company, constantly increasing the scope of it's projects, it's capabilities, and it's vision. With that comes the amazing opportunities that only comes from a small/medium sized company! There aren't many companies where you can have something you designed or built in space within 1~2 years! If flying something you've worked on in space is a career goal of yours, I can't think of a better place to be. Another side effect of a rapidly growing company is the opoprtunities for increased responsibility. SpaceX employees often find themselves responsible for very important elements very quickly. Responsibility it placed upon people based more on merit than tenure, which allows for career growth not possible with larger aerospace companes. In my experience, SpaceX does a good job of rewarding efforts and successes with perks and bonuses. While compensations may be below aerospace industry standards initially, SpaceX (in my opinion) does a better job of rewarding performance. At the end of the day, there isn't any thing quite like watching a rocket take off and a spacecraft at the space stations, and SpaceX captures that excitement and passion of that dream!
Cons
Hours and work/life balance at SpaceX is likely the biggest and most-common complaint, but in my experience it is what each person wants to make of it. SpaceX will take as much as you want to give it, which for a lot of energetic fresh from college engineers is alot. There are lots of folks who have made a successful transition from 60+ hour workweeks to something more sustainable, but you must make a concious decision to do so. For better or worse, there is definitely less structure, organization, and training at SpaceX than other companies. Nearly all of the training and learning is "on the job" training, as there are minimal mentoring programs. Also, some of the institutional protections of larger companies will not there at SpaceX (for better or worse).