The main advantage of doing the ABA degree first (assuming it makes you eligible for the BCBA certification) is that you will have a highly marketable credential (with starting salaries similar to doctoral level psychologists in many areas) that you can use to "subsidize" your psych training- or as a backup if the psych thing doesn't work out. With new insurance mandates in many states to cover autism services, as well as increasing rates of ASD diagnoses, the demand for BCBAs is much larger than the supply in many areas.
The advantage of doing it the other way around (e.g. masters/PhD in psych first) is that it can cut down on the overall amount of coursework. Much (most?) of your ABA masters coursework won't count towards your psych degree (ABA has really become a distinct discipline, rather than a branch of psychology). You'll take the required 6 courses for the BCBA certification, plus others required for the masters- often education/special education courses). However, if you have an existing masters or higher degree in a related field (psych still counts), you can be eligible for the BCBA credential by taking only the 6 course sequence (for definitive and ever changing requirements, consult the BACB.com website, rather than rely on the word of some random guy on a web forum).
To correct an item from a post above- a BCBA-D (doctoral level BCBA) does not really confer any benefits, as a credential, above a BCBA, but rather represents a different degree of training and scholarship. It does not allow you teach. I have a BCBA-D and teach in ABA graduate programs. It is my PhD that allows me to teach at the professor level (plus experience, knowing the right people, and being in the right place at the right time). There are occasionally masters level instructors as well.
The combo psych licensure/BCBA credential can be desirable and open up many options for you (or the agency you work for). In my current position, I split my time between being a "psychologist" and doing diagnostic evals with kids under 3 displaying potential autism spectrum disorder (billing insurance and other payers under my psych license) and being a "behavior analyst" and designing and supervising home-based ABA programs, parent training, etc. (billing insurance and other payors under my BCBA credential). It's a great combo of credentials, allowing for an interesting and varied job where I get to do a lot of things.