In this paper, we further investigate the constructions of complementary sequence sets (CSSs) and complete complementary codes (CCCs) by Butson-type Hadamard matrices. By taking the algebraic structure of Butson-type Hadamard (BH) matrices into consideration, we obtain the explicit representation of the δ-linear terms and δ-quadratic terms, which are ingredients to construct CSSs and CCCs. In particular, we derive the δ-quadratic terms determined by DFT matrices and BH matrices constructed from 2-level autocorrelation sequences, which yields two type of new contructions. We show that inequivalent BH matrices produce different CSSs and CCCs, which proves that our constructed CSSs and CCCs are new. As a consequence of the first type of the constructions, not only a large number of p -ary CSSs and CCCs of size p ( p prime) have been proposed, which were never reported in the literature, but also a theory linking these CSSs of p -ary sequences and the generalized Reed-Muller codes proposed by Kasami et al . is shown. These codes enjoy good error-correcting capability, tightly controlled PMEPR, and significantly extend the range of coding options for applications of OFDM using p n subcarriers. As a consequence of the second type of the constructions, we reveal an extremely fascinating hidden connection between the sequences in aperiodic CSSs and CCCs and the sequences with ideal period 2-level autocorrelation, through their trace representations and permutation polynomials over finite fields.