What are the pros and cons to having a 1 or 2 part implant (i look at the implant crown as being the 3rd part)? Cost difference, etc.?

Outcome, forget cost. In my opinion, except in very straightforward cases, it is important for both the restorative dentist & the surgeon to work together in planning the case from start to finish. When the abutment is placed at the time of surgery, you are stuck with the standard abutment "guestimate". When the implant is fabricated after the fixture is placed, a custom abutment can fabricated from an impression.
Several. Traditionally, most implants are done as you state in three parts, the implant, the abutment which connects into the implant and supports the crown. These are more customizable and preferable. Generally single stage implants which incorporate both the implant and abutment are used in very narrow situations only.
Predictability. The big difference is the success rate of a one stage vs. A two stage procedure. Depending upon factors such as bruxism or clenching, tongue thrust or the number of remaining teeth this will impact upon the success of the dental implant. The more stress or strain placed on the implant by these factors may jeordize the success of the dental implant.
There should be no . Cost difference between single stage and two stage implant surgery. Single stage (healing abutment and implant are placed same time eliminating second soft tissue uncovering for placement of healing abutment later) two-stage, the implant has cover screw placed and gingiva sutured over, requiring an uncovering later. Some situations dictate 2 stage, while most circumstances allow for 1 stage.
Implant crown. Hi, good question. Single piece implant: adv. Less components (less costly), less risk for abutment to come loose (no middle part), potentially less bone loss at the collar level (bone level) disadv: no flexibility as a restorative option. If the implant placement is off, no way to compensate for it with a custom abutment. No porcelain abutments for front teeth, rendering less translucent crowns.
No cost difference. A single implant tooth replacement for 1 missing tooth is composed of 3 parts: the implant (replaces tooth root), the abutment (connects implant to prosthetic), and crown/prosthetic. This being said, some implant restorations have only 2 pieces--in this case, the abutment is a part of the implant and the prosthetic is screwed directly into in implant. There is no price benefit. Tx benefit only.