What are the main differences between $p p$ and $p \bar p$ colliders I know that it is somehow related to the parton distribution functions, allowing specific reactions with gluons instead of quarks and anti-quarks, but I would really appreciate more detailed answers !
Thanks
 A: I would add to @anna's answer that $p\bar{p}$ collider such as the Tevatron is CP symmetric. This was one of the arguments for continuing the Tevatron Experiments. To quote from the proposal

Measurements that get a special
  advantage from the p-pbar environment.
  The  primary example in this category
  is CP-violation, which strongly limits
  the range of allowed  models of new
  physics up to scales of several TeV.
  There are good a priori reasons to
  expect the  existence of some non-SM
  CP-violating processes, and finding
  them is of comparable  importance to
  addressing electroweak symmetry
  breaking. Precision measurements at
  the 1%  level or better are accessible
  at the Tevatron due to the
  CP-symmetric initial state (p-pbar),
  and  symmetry of the detectors that
  allow cancellation of systematics.
  Some of these measurements  already
  show tantalizing effects, like the
  recently published di-muon asymmetry
  result from the  DZero experiment,
  showing the first indication of a
  deviation from the Standard Model
  picture  of CP-violation. Other
  measurements are exploring a
  completely new field, as the recent
  CPVmeasurement with the D 0 mesons at
  CDF, yielding a substantial
  improvement in precision with  respect
  to previous B-factories data. This has
  provided a proof of feasibility of an
  exciting program of precision
  measurement with a unique possibility
  to find anomalous interactions in 
  up-type quarks. A non CP-related
  example in this category is the
  forward-backward asymmetry  in top
  quark production. Current measurements
  by both CDF and DZero indicate an
  asymmetry  above the Standard Model
  prediction. If this persists with more
  data, it can be interpreted as  new
  dynamics. This is not an easy
  measurement to replicate in a
  proton-proton environment.

http://www.fnal.gov/directorate/Tevatron/Tevatron_whitepaper.pdf
A: The difference  in scattering cross sections is more evident the lower the energy of collisions. Fig 41.11. At the energies of TeV the probability of new physics observations is the same for both choices of collisions.
The reason is that at low energies the fact that the proton has three quarks and the anti proton three anti quarks predominates. Quark antiquark scattering at low energies has much higher cross section than quark quark due to the extra possibility of annihilation of the quarks. At low energy the gluon "sea" plays a small part. The higher the energy of interactions the higher the number of energetic gluons that scatter and finally at TEV energies that is what predominates and the two cross sections converge.      Thus for physics it makes no difference whether one uses as targets protons or antiproitons, as far as discovery potential goes.
There may be some technical advantage in the construction, in that in principle the antiproton-proton beams can circulate in the same magnetic configuration as mirror images and make the magnet construction circuits simpler. I guess that the need for high luminosity made LHC a proton proton collider, since it is more difficult to store antiprotons. I would have to research this guess.
A: From the machine side, a symmetric $p\bar{p}$ collider can have only one beampipe, so it is much simpler. On the other hand if you fill it with many bunches they will start to collide all around the machine. You may manage to separate their orbits, but they will still feel the fields reciprocally generated (long-range beam-beam interaction) that will limit the beam intensity. So, even if you could produce an arbitrary amount of $\bar{p}$ (which would still be a major limitation), you won't be able to fill the machine with a very high current.
With two separate beam pipes this problem is limited to small sections close to the interaction regions. Two rings allows you also to better optimise each beam. In the end you will achieve an higher maximum current and so luminosity. It is also possible to store the same species but also totally different species, like protons and lead ions. The price to pay is a much more expensive and complicated machine in which many system (and so the failure probability/downtime) are replicated twice.
