Both OFDM and single-carrier block transmission (SCBT) are promising candidates for the future high date rate communication, which well handle the frequency selective channels with a low implementation complexity. However, the sensitivity to carrier frequency offset, caused by Doppler effect or the carrier frequency difference between transmitter and receiver, is always a major concern of these block transmission systems. Actually, a controversy in comparing the carrier frequency offset effect on OFDM and SCBT systems has already been witnessed in the literature for a long time. In this paper, we first analyze the effects of carrier frequency offset on these two systems, then by carefully investigating existing works, we classify the problem into two categories: carrier frequency offset analysis with compensation and without compensation. Finally, we draw conclusions based on theoretical and numerical results that the BER performance of OFDM systems is more sensitive to carrier frequency offset than SCBT systems for the BPSK modulation in AWGN channels, although the difference of the BER performance between two systems is much smaller than theoretically predicted in previous works, if a fair comparison environment is used. Moreover, this small performance gap vanishes in higher level constellation like QPSK.