Hi,
They are two different programs.
Acrobat
This can be used to generate native PDFs from other document types. In addition you can create forms that are called AcroForms. These retain most of the functionality that is in Acrobat, like commenting.
The form IS a PDF.
Acrobat is avialable for Windows and Mac.
LiveCycle Designer
This program is used to develop forms and is completely separate to Acrobat. The forms are called XFA Forms, which stands for XML Forms Acrhitecture. The focus here is very much on data processing.
While forms that are created in LC Designer can be opened in Acrobat (and Reader), they are not native PDFs and some of the normaly functionality is not available. For example, while you can add data to fields when the form is opened in Acrobat, you cannot edit the form itself in Acrobat. If you want to edit the form, you would need to open it again in LC Designer.
It comes packaged with the Windows version of Acrobat Pro. There isn't a Mac version of LC Designer.
That is a fairly broad brush outline, hope it helps,
Niall