Hi, I'd like to help you with your question.
Bipolar disorder is characterized by depression (sadness, trouble with sleep and appetite, possible suicidal feelings) and mania (extreme feelings of wellbeing, excitability, grandiose beliefs, etc). It has many subcategories to address the different ways the depression and mania express themselves.
Borderline personality disorder is characterized by instability with interpersonal relationships, chronic feelings of emptiness, suicidal gestures, instability in mood.
Bipolar disorder is considered an Axis I disorder, a major mental health disorder. Borderline is considered an Axis II disorder, which is a secondary disorder.
The similarity between the disorders is the mood instability. Both have shifts in mood. However, with Bipolar the mood lasts longer than with Borderline, which is a more unstable mood shift. The mood shifts with borderline disorder can be as a result of situational changes (trouble with a friend, bad day at work) and the change in mood does not include feelings of mania, whereas Bipolar tends to come on suddenly and includes the mania.
You can be diagnosed with both disorders at the same time. That may be a possibility, and it is not uncommon to have an Axis I and Axis II diagnosis. Ask your doctor if this is possible for you. However, if you are unsure about your current doctor's diagnosis or he/she is not willing to reevaluate you, it never hurts to obtain a second opinion. Express your concerns to the evaluator and they will help you sort out your symptoms for a more accurate diagnosis or at least confirm the one you have.
I hope this has helped you,