They are often used interchangeably. However, a "part" is often an area of the whole that can somehow be identified as unique -- as having it's own smaller function. For example, body "parts" like legs and arms and ears...they are all different from each other and do different things. A "section" would more often be used when the whole area is divided up into smaller areas geographically, but each area does not have it's own specific purpose and is divided more for convenience. For example, this "section" of the cake is mine, or this "section" of the country tends to vote for Republicans.

It depends on the context.