What is the difference between Distant (D) NOTAMs, Local (L) NOTAMs and FDC NOTAMs in the US?
Short answer: D NOTAMs are about airports and navigation facilities; FDC NOTAMs are about airspace, procedures or regulations; and L NOTAMs don't exist any more (unless you're in the military).



The information on NOTAM types is a bit messy and inconsistent, to say the least. The Alaskan FSS lists 11 types; the PHAK 13-3 lists 3 types (as acpilot mentioned in his answer); the AIM 5-1-3 lists five in its NOTAM section, plus GPS NOTAMs in the GPS section.

And the FAA's military NOTAM site lists yet more types in their FAQ  such as M, V, W and F.

Anyway, here's what the AIM says about D and FDC NOTAMs:


  NOTAM (D) information is disseminated for all navigational facilities
  that are part of the National Airspace System (NAS), all public use
  airports, seaplane bases, and heliports listed in the Chart Supplement
  U.S. [...]  NOTAM (D) information includes such data as taxiway
  closures, personnel and equipment near or crossing runways, and
  airport lighting aids that do not affect instrument approach criteria,
  such as VASI.
  
  [...]
  
  FDC NOTAMs. On those occasions when it becomes necessary to
  disseminate information which is regulatory in nature, the National
  Flight Data Center (NFDC), in Washington, DC, will issue an FDC NOTAM.
  FDC NOTAMs contain such things as amendments to published IAPs and
  other current aeronautical charts. They are also used to advertise
  temporary flight restrictions caused by such things as natural
  disasters or large-scale public events that may generate a congestion
  of air traffic over a site.


According to the Alaska FSS page, NOTAM(L) doesn't exist any more for civilian use:


  NOTAM (L):  A NOTAM given local dissemination by voice and other
  means, such as telautograph and telephone, to satisfy local user
  requirements.  This type of NOTAM is now used only in the military
  NOTAM system.  All NOTAMs previously considered NOTAM (L)s in the
  civil NOTAM system are now considered NOTAM (D)s.


That's confirmed by AIR FORCE INSTRUCTION 11-208 (I), which describes NOTAM(L) as a military type:


  4.2.1. Military.
  [...]
  4.2.1.2. Local NOTAM (L series). Local NOTAMs are non-critical and non-safety related information that is essential for the flying and
  support communities to know. Facility managers may establish the
  parameters for local NOTAM determination.


Civilian L NOTAMs are now D NOTAMs that start with the specific facility code. You can find them on the civilian NOTAM site, e.g.:


  !LOU 05/077 LOU RWY 15/33 CLSD
  1605091213-1606222000


And to address your final point, NOTAMs are not obsolete. There needs to be some way to inform pilots about temporary changes or things that happen at short notice, and that's what NOTAMs are for. Many people feel that the system could be improved somehow - pulling up the NOTAMs for even a short flight can give you dozens of results - but there's no doubt that it's needed. Reviewing NOTAMs is an important preflight step, and examiners often test you on it during checkrides. There are plenty of NTSB accident reports that mention NOTAMs, usually in cases where the pilot didn't review them.