Javier, try looking at What is an intuitive picture of the motion of nucleons? to start.
The short answer is that it is as reasonable to say that they are identical as it is to say that the configuration of electrons in multiple atoms of a single element are identical. That is, there is a set of position (or momentum1) distributions to which they conform.
1 The position and momentum distributions turn out to be linked to each other by Fourier transformations, so information required to specify one is the same as that required to specify the other. Nuclear physicist mostly concern themselves with the momentum distributions.
Protons and neutrons are referred to collectively as nucleons. Nucleons interact via the strong nuclear force, and this interaction can't be expressed by any simple equation. The reason is that nucleons are not fundamental particles. They're actually clusters of quarks.
Short range
The low-energy structure of nuclei is amazingly insensitive to the details of the nucleon-nucleon interaction that you pick as an approximation to the actual underlying quark-quark interaction. This is both good and bad. It's good because you don't need to understand very much about the nasty details in order to find out the properties of nuclei, e.g., why nuclei have the sizes and shapes they do. It's bad because it means that you can never infer very much about the interaction simply by observing properties of nuclei. As an example of how insensitive nuclear structure is to the details of the strong nuclear force, clusters of sodium atoms have magic numbers that match up with the first few magic numbers for nuclei; this is because these magic numbers only depend on the short-range nature of the interaction.
Other effects that can be understood based on the short range of the interaction are:
Nuclei act as though they have surface tension (so they resist being deformed).
Nuclei are most stable if they have even numbers of neutrons and even numbers of protons (because then the neutrons and protons can pair off in time-reversed orbits that maximize their spatial overlap).
Nucleons in an open shell tend to couple so as to form the minimum total angular momentum (the opposite of Hund's rules for electrons).
A residual interaction
The short-range nature of the nuclear interaction is very surprising, because the quark-quark force is believed to be roughly independent of distance. What's happening here is that nucleons are color-neutral, just as a hydrogen atom is charge-neutral. Just as hydrogen atoms "shouldn't" interact, nucleons "shouldn't" interact either. The forces between nucleons very nearly cancel out, and likewise the electrical forces between two neutral hydrogen atoms very nearly cancel out. The nonvanishing interaction comes from effects like the polarization of one particle by the other. For this reason, the nucleon-nucleon interaction is referred to as a residual interaction.
Other than its coupling constant and its range, what other features of the nuclear interaction are important for understanding low-energy nuclear structure?
Spin-orbit
There is a spin-orbit interaction, which is much stronger than, and in the opposite direction compared to, the one expected from special relativity alone.
Symmetry between neutrons and protons
The nuclear interaction remains unchanged when we transform neutrons to protons and protons to neutrons. For this reason light nuclei exhibit nearly identical properties if you swap their N and Z. Heavy nuclei don't have this symmetry, which is broken by the electrical interaction.
No qualitative features inferrable from sizes of nuclei
We do not get any clearcut, qualitative information about the interaction based on the observed sizes of nuclei. An extremely broad class of interactions between point particles results in n-body systems that have bound states and finite density. The finite density (i.e., the lack of a total collapse to a point) is essentially a generic result of the Heisenberg uncertainty principle. Only for certain special types of potentials that blow up to $-\infty$ at short ranges can one circumvent this (Lieb 1976).
A variety of models
Because the nucleon-nucleon interaction is a residual interaction, and nucleons are really composite rather than pointlike, the whole notion of a nucleon-nucleon interaction is an approximation, and one can model it in a variety of ways while still producing agreement with the data. In particular, some models have a hard, repulsive core, while others do not,(Chamel 2010, Stone 2006) and both types can reproduce the observed sizes of nuclei. This disproves the common misconception that such a hard core is needed in order to explain the sizes of nuclei.
References
Chamel and Pearson, 2010, "The Skyrme-Hartree-Fock-Bogoliubov method: its application to finite nuclei and neutron-star crusts," http://arxiv.org/abs/1001.5377
Lieb, Rev Mod Phys 48 (1976) 553, available at http://www.pas.rochester.edu/~rajeev/phy246/lieb.pdf
Stone and Reinhard, 2006, "The Skyrme Interaction in finite nuclei and nuclear matter," http://arxiv.org/abs/nucl-th/0607002
Best Answer
Evidence that there are distinct protons and neutrons in nuclei starts with the Pauli term (pairing term) in the semiempirical mass formula of the liquid drop model.
Furthermore, all nuclei with even numbers of protons and neutrons have nuclear spin of zero. This is consisent with shells being filled with spin up and spin down pairs of nucleons, each pair resulting in net zero spin.
More generally, that experimental data are consistent with the Nuclear Shell Model is evidence that distinct protons and neutrons exist in the nucleus.
Also, the protons and neutrons are held together by exchange of pions. The exchange can result in the proton becoming a neutron and a neutron becoming a proton, so it is not that they exist entirely "as is".
See A reappraisal of the mechanism of pion exchange and its implications for the teaching of particle physics for furthur discussion of pion exchange.