Transmutation of the legacy TRansUranics (TRU) from Light Water Reactor operation has become in recent years a main objective for the development of Fast Reactors (FR). In fact, an effective TRU-burning requires fuel multi-recycling and a fast-neutron-spectrum reduces the endogenous generation of Cm and Cf isotopes, thus benefitting fuel handling and in-core radiotoxicity generation. However, achievement of high TRU-burning rates requires low-Conversion-Ratio (CR) reactors with a high fraction of Minor Actinides (MA) in the core, requiring remote fuel fabrication behind thick shielding. Problems of fuel handling are exacerbated if Th is used as fertile isotope (e.g. to enhance safety or TRU-burning rate), since Th-232 irradiation causes the build-up of U-232, whose progeny emits high energy gamma rays. Use of a liquid fuel with online reprocessing would avoid most of the issues related to reprocessing, manufacturing and transporting highly radioactive recycled fuel. The logical technology for the adoption of liquid fuel is the Molten Salt Reactor (MSR). Among MSRs, the Molten Salt Fast Reactor (MSFR) is in principle better suited for TRU burning as it combines the advantages of a liquid fuel with those of a fast-spectrum and of Th use. Objective of this work is to evaluate the MSFR potential benefits in terms of TRU burning through a comparative analysis with a sodium-cooled FR. The comparison is based on TRU- and MA-burning rates, as well as on the in-core evolution of radiotoxicity and decay heat. Solubility issues limit the TRU-burning rate to 1/3 that achievable in traditional low-CR FRs. The softer spectrum also determines notable radiotoxicity and decay heat of the equilibrium actinide inventory. On the other hand, the liquid fuel suggests the possibility of using a Pu-free feed composed only of Th and MA, thus maximizing the MA burning rate. This is generally not possible in traditional low-CR FRs due to safety deterioration and decay heat of reprocessed fuel. In addition, the high specific power and the lack of out-of-core cooling times foster a quick transition toward equilibrium, which improves the MSFR capability to burn an initial fissile loading, and makes the MSFR a promising system for a quick (i.e., in a reactor lifetime) transition from the current U-based fuel cycle to a novel closed Th cycle.