You might find this recent thread useful:
I want to be an IBCLC.
No actual IBCLCs (I don't think) have posted there - it would be great to hear from them as to how they got there.
Basically there are now three paths one can take to qualify to take the IBCLE exam:
1. Healthcare Professionals and trained Breastfeeding Supporters (e.g. LLL leaders and the like), whether voluntary or professional
2. People who have comprehensively studied lactation in an academic institution
3. People who have worked as an "intern" (basically) under the supervision of an experienced IBCLC
Most people worldwide take pathway one.
More information on the pathways
here - click on the drop down menu to get info on each of the three pathways.
As for the difference between the two acronyms, my understanding is that the term "Lactation Consultant" is not a protected term, therefore anyone can call themselves an LC, and any training insitution can offer certificates certifying Lactation Consultants. I understand that the International Board exam is the gold standard.