Atserias, Kolaitis, and Vardi showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD(∧, weakening), simulates CP^∗ (Cutting Planes with unary coefficients). We show that OBDD(∧, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring tautologies have polynomial size proofs in the OBDD(∧, weakening) system. The reordering rule allows changing the variable order for OBDDs. We show that OBDD (∧, weakening, reordering) is strictly stronger than OBDD(∧, weakening). This is proved using the Clique-Coloring tautologies, and by transforming tautologies using coded permutations and orification. We also give CNF formulas which have polynomial size OBDD(∧) proofs but require superpolynomial (actually, quasipolynomial size) resolution proofs, and thus we partially resolve an open question proposed by Groote and Zantema. Applying dag-like and tree-like lifting techniques to the mentioned results, we completely analyze which of the systems among CP^∗, OBDD(∧), OBDD(∧, reordering), OBDD(∧, weakening) and OBDD (∧, weakening, reordering) polynomially simulate each other. For dag-like proof systems, some of our separations are quasipolynomial and some are exponential; for tree-like systems, all of our separations are exponential.