Answer:
   
   Both agree that Chometz products are forbidden. Ashkenazi authorities
   additionally forbade kitniyos, a class of foods in some ways similar
   to chometz, but not classified as "chometz." Kitniyos refers to grains
   and grain like products such as rice, millet, beans, lentils, and
   others. Even though these items cannot become chometz, Ashkenazim do
   not eat them because they are easily confused with grains that can be
   become chometz and may even be mixed together with them. Sephardic
   Jews (Jews from primarily the Middle East and Northern Africa)
   generally do not refrain from eating kitniyos. Possession of kitniyos
   is permitted according to all customs.
   
   The custom of avoiding kitnyos is mentioned for the first time in
   France and Provence in the beginning of the thirteenth century by R.
   Asher of Lunel; R. Samuel of Falaise, and R. Peretz of Corbeil - from
   there it spread to various countries and the list of prohibited foods
   continued to expand. Nevertheless, the reason for the custom was
   unknown and as a result many sages invented at least eleven different
   explanations for the custom. The most common explanation appears to be
   that kitnyos grains may be ground and look like flour, and that the
   swell in contact with water. Thus, to avoid confusion, Ashkenazi Jews
   avoided them.
   
   There is a long discussion of the origins of the customs and its
   specifics at [5]http://www.tzemachdovid.org/klh/taubes.html.
   
   For Ashkenazi Conservative Jews, the Conservative Movement has issued
   a tshuva stating that kitnyos may be eaten on Pesach. It can be found
   at [6]http://www.jtsa.edu/org/masorti/msg00085.html, and states that
   the custom of kitnyos is in direct contradiction to an explicit
   decision in the Babylonian Talmud (Pesachim 114b), as well as the
   opinion of all the sages of the Mishna and Talmud except one (R.
   Yochanan ben Nuri, Pesachim 35a and parallels). The Tshuvah also
   claims that it contradicts the theory and the practice of the Amoraim
   both in Babylonia and in Israel (Pesachim 114b and other sources), the
   Geonim (Sheiltot. Halakhot Pesukot,,Halaktiot Gedolot, etc.) and of
   most of the early medieval authorities in all countries (altogether
   more than 50 Rishonim!).