Positives:
- Willingness to try new approaches like the reusable rocket stages of SpaceX
- Less chance of Congress keeping useless initiatives alive just because of which districts they are located in (look at SLS)
- Less expensive access to space
Negatives
- As Jim pointed out, Elon's microsats could be a real problem for earth-based observatories and eventually for other satellites
- Safety will undoubtedly be compromised in human spaceflight. NASA is extremely cautious.
- I don't think private spaceflight will advance pure research other than perhaps making a launch less expensive for Mars probes, etc. NASA does an amazing job in scisnce missions.