$n$-body problem = many-body problem? Are the terms "$n$-body problem" and "many-body problem" synonymous? Or does one refer to a numerical problem an the other to an analytical problem?
 A: N-body problem refers to the problem of having any number objects with initial positions and velocity and predict their dynamics according to Newton's laws; for n = 2 there are analytical and practical solutions, and though it is proven that analytic solutions exists for n > 3 those are not practical and numerical methods are used instead.
Many-body problems refer to the family of problems pertaining a quantum system with an arbitrary number of particles.
A: The Mathematics Subject Classification (MSC2010) classifies "many-body" under quantum mechanics:

81-XX Quantum
  theory 149997
     81Vxx Applications to specific physical systems 33023
          81V70 Many-body theory; quantum Hall effect
4367

Whereas "$n$-body" appears to have more extension:

70-XX Mechanics of
  particles and systems
  [For relativistic mechanics, see
  83A05 and
  83C10; for
  statistical mechanics, see
  82-XX]
  62456
     70Fxx
Dynamics of a system of particles, including celestial
  mechanics
14431
          70F10
$n$-body problems
2495
81-XX Quantum
  theory
149997
     81Uxx
Scattering theory
  [See also 34A55,
  34L25,
  34L40,
  35P25,
  47A40]
  8728
          81U10
$n$-body potential scattering
  theory
850

APS's Physics Subject Headings doesn't appear to have "$n$-body," nor does AAS's Astronomical Subject Keywords even have "many-body".
The OED doesn't have "$n$-body", but it says this for "many-body":

many-body adj. relating to or involving three or more bodies or particles; spec. with reference to the problem of predicting their positions and motions at any future time given their present values and the way the bodies interact (through collision, gravitational attraction, etc.).

