The entanglement of a two dimensional particle and the one of two one dimensional particles 
Case 1:
  Usually, entanglement is a correlation between more than one particles. For example, the momentum $p_1$ of the particle $1$ and the one $p_2$ of the particle $2$ have the relation that $p_1 + p_2 = 0$.



Case 2: Similarly, one can also think about the entanglement between different dimensions of a particle. For example, the momentum $p_x$ in the $x$ direction and the one $p_y$ in the $y$ direction have the relation that $p_x + p_y = 0$.

Then the question is:


*

*Is there any difference in the mathematical description of the entanglement in these two cases?

*Is there any difference in the physical description of the entanglement in these two cases?
 A: Mathematically there is no difference between entanglement between degrees of
freedom in two different systems or particles and between degrees of freedom of
the same system or particle.
Entanglement between two degrees of freedom in the same system/particle is
sometimes called "classical entanglement," due to the fact that such
correlations can be demonstrated in a coherent, yet classical system, such as
with a laser beam (radial polarization is an example of a classical laser
beam where the polarization and spatial degrees of freedom are "entangled").
For reference, two of the major papers on the field of classical entanglement
are listed below.
Although the two types of entanglement you list seem mathematically similar,
there is a physical difference. Specifically you can't separate the different
entangled degrees of freedom if they are part of the same system/particle, and
so you can never demonstrate any type of non-local correlations (e.g. via Bell
violations). This has for using this type of entanglement as a resource,
for example you lose the speedup that is the advantage in quantum computing for
these types of systems, as Ref [1] below states

The same issue, the fact that the two cebits (classical qubits) cannot be
  spatially separated, provides a reason why it is impossible to use the cebits
  as qubits and thus build a quantum computer. For example, one could not build
  up a network of quantum logic gates, whereby two output ports of one gate are
  sent into the inputs of two other gates. Nonlocality thus appears to be an
  essential ingredient in the functioning of quantum logic.

References:


*

*K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh,
"Bell’s measure in classical optical coherence," Nat. Photonics 7, 72–78
(2012) http://dx.doi.org/10.1038/nphoton.2012.312.

*R. J. C. Spreeuw, "A Classical Analogy of Entanglement," Found. Phys. 28,
361–374 (1998) http://dx.doi.org/10.1023/A:1018703709245. pdf online here.

