Scattering of energetic electron and proton beams by cold matter is significantly different from the scattering of these particles by plasma, which may be either highly ionized or dense strongly coupled plasma. This is due to the difference in the shielding of the target nuclei between the two cases. Quantitatively, we treat the problem by means of the Bethe Moliere multiple scattering theory and the version of this theory for plasma as derived by Lampe. We propose to use this effect as a plasma diagnostic tool, utilizing monoenergetic, well-collimated electron or proton beams produced either by femtosecond laser plasma interactions or by accelerators. The effect is first illustrated for simplicity, by calculating the widths of the angular distribution of scattered particles interacting with the extreme cases of very hot fully ionized carbon, and iron plasmas, and comparing these results to the corresponding cold material. The more relevant case of electron scattering from partially ionized iron and carbon plasmas covering the entire range from a cold to a completely ionized target is also dealt with here. This paper brings up and highlights the difference between scattering by plasma and by cold material in light of the recent proposals to employ particle beams for various fusion applications.