@umanbean pretty much nailed it in a very simplistic way.
Keep these points in mind:
A) Hobbyist "notifies" tower or controlling agency(
tower, airport manager etc) if flying within 5 miles of an airport (
airspace doesn't matter except Class B in which they must get permission before entering into BRAVO). Except as noted above in
BRAVO, you are merely notifying of your intentions and not asking permission but the agency CAN deny and insist you not fly. If you go against that you could be facing local, state and Federal repercussions so that's a choice you and only you can make. They are only supposed to deny the flight if it causes a threat to NAS in some way. If it truly does and you fly any way don't be surprised if you're met with local authorities and cited or more.
B) Part 107 doesn't contact the tower to fly at a controlled airport. They have to get Airspace Approval via the online web portal which could take 90+ days so plan way ahead. Some of the approval "Authorizations/Waivers" do require tower notification 30min prior to the flight but that is clearly spelled out in each individual authorization/waiver.
C) Part 107 can fly under Part 107 or Part 101 (hobby) rules but whatever you take off flying under the entire flight must be flown according to that designation. You can NOT mix & match the two in flight.
D) Part 107 can fly as hobbyist but hobbyist can NOT fly as Part 107.
E) As of this writing, December 12th, 2017 8:12am (EST) hobby aircraft are NOT required to be registered with the FAA. This is about to change any day now so be sure to research thoroughly if you are new or deciding to register or not. Personally I think ALL sUAS over .55lbs SHOULD be registered but that's my 2 cents.
F) A sUAS that is being flown under Part 107 is legally required to be registered with the FAA as a Commercial Aircraft and identified as such.
G) An aircraft registered with the FAA as a Commercial Aircraft CAN be flown for hobby/recreation with no additional registration/processing required.
H) An aircraft registered with the FAA as a HOBBY Aircraft (or unregistered) CAN be flown for Hobby and ONLY Hobby flights.
I) Part 101 (hobby) UAS operations are a subset of Part 107 Operations. If any portion of your flight does not fit perfectly within the definition of Part 101 your entire flight defaults to Part 107 Rules and are enforceable as such.