This paper studies the reconfigurable intelligent surface (RIS)-assisted symbiotic radio (SR) system, where an RIS acts as a secondary transmitter to transmit its information by leveraging the primary signal as its RF carrier and simultaneously assists the primary transmission. Conventionally, all reflecting elements of the RIS are used to transmit the secondary signal, which, however, would limit its capability for assisting the primary transmission. To address this issue, we propose a novel RIS partitioning scheme, where the RIS is partitioned into two sub-surfaces, one to assist the primary transmission and the other to transmit the secondary signal. Naturally, there exists a fundamental tradeoff between the assistance and transmission capabilities of RIS regarding the surface partitioning strategy. Considering the coupling effect between the primary and secondary transmissions, we focus on the detection of the composite signal formed by the primary and secondary ones, based on which we propose a novel two-step detector. Then, we formulate the assistance-transmission tradeoff problem to minimize the bit error rate (BER) of the composite signal by jointly optimizing the surface partitioning strategy and the phase shifts of the two sub-surfaces, such that the overall BER of RIS-assisted SR is minimized. By solving this problem, we show that the optimized surface partitioning strategy depends on the channel strength ratio of the direct link to the reflected link. Finally, extensive simulations show that our proposed RIS partitioning scheme outperforms the conventional schemes which use all reflecting elements for either assistance or transmission.