The current thinking is that Borderline personality disorder (BPD) and bipolar are part of the same continuum, just different severity. Basically that BPD is the weakest variant and BP I is the strongest.

However, by DSM IV standards, one is considered an axis I mood disorder, and the other is considered an axis II personality disorder (and therefore much harder to treat). Borderline has a very negative reputation among mental health professionals, as these people are often very dramatic and difficult to help.

My personal opinion is that borderline and bipolar are not variants of the same continuum because the prevalence of bipolar is even in both sexes, whereas borderline occurs MUCH more commonly in women. Also, the key features of the two are quite different (in my opinion) - for instance, bipolar is not generally associated with self-harm behaviors, whereas that is a common and almost pathonomonic feature of borderline.

It's complicated, but I recommend that you not push for the borderline diagnosis. Providers, unfortunately, WILL treat you differently, and will likely write you off as being a hopeless drama queen, and not a helpable person. For the purposes of getting treatment, it's better to be bipolar.