Pros
You get to work on amazing projects. The staff is brilliant and the culture is family-like and supportive. People work hard and have fun at the same time. The exhibits and programs are world class at best, but sometimes can be spotty. The work environment is very unstructured and there are always interesting things going on around you.
Cons
Dysfunctional senior management and lack of organizational structure. There are many competent, well-meaning managers, but they don't have the resources or training to support their staff properly. Poor financial planning and oversight lead to semi-regular budget crises. Those, in turn, lead to recurring layoffs and destructive and unexpected budget cuts. The downside to the family-like environment is cliqueness and cronyism. Promotions and resources are not always allocated fairly. Poor communication by senior management to rank-and-file staff leads to lack of trust. The institution is not as egalitarian as it thinks it is, there is an unspoken caste system: some people get known as "makers" and get to contribute creatively, others are left out; it's hard to move from one group to another. This is in direct contrast to the outward mission of the institution. There is a sincere push by management to address some of these issues but only time will tell whether it is effective or not.