Pros
Great place to learn, especially early in your career. The firm has a small firm, entrepreneurial feel, which is a unique cultural point for BX. Thus, you get a lot of exposure to senior people and you can see the impact of your work (if you're on the investment, portfolio operations, or IRBD side of things - not sure about back office).
Cons
Culture is very team-specific, so do your due diligence. People can be passive aggressive or just plain aggressive. There's a lack of focus on training mid-level (VP and above) employees and there is a pretty wide array of competency in managers, from a pure managerial perspective. Also beware of growth trajectory of different business lines/business units and how the business unit has been doing.