The Cantonese/Taishanese Initial Mapping Table is reproduced with asterisks added to the first column right next to the Consonant Initial Numbers for mappings that are significantly different.

            Cantonese Initials 广州话声母                      Taishanese Initials 台山话声母
1 b [p] b [p]
2 p [ph] p [ph]
3 m [m] m [m]
4 f [f] f [f]
5* d [t] null [?]
6* t [th] h [h]
7 n [n] n [n]
8 l [l] l [l]
9 g [k] g [k]
10 k [kh] k [kh]
11 ng [ŋ] ng [ŋ]
12 h [h] h [h]
13 gw [kw] g [k]
14 kw [kwh] k [kh]
15 w [w] v [v]
16* z [ts] 制, 祭 dz, d [ts], [t]
17* c [tsh] 產, 餐 ts, t [tsh], [th]
18* s [s] 勢, 細 s, ɬ [s], [ɬ]
19* j [j] 鹽, 嚴 y, ŋ [j], [ŋ]


It seems like the mismatch can be accounted for by three chain shifts in Taishanese triggered by the mergers of affricate and fricative consonant initials.
The mergers took place in both dialects. However, in Taishanese each merger also triggered a corresponding chain shift. Below is a summary of the proposed chain shift model:
        L1   L2   L3  
1.    / tɕ  (莊/章) ts  (精) t  (端) ?
2. h / h(初/昌) tsh(清) th(透) h
3.         ʃ   (生) s (心) ɬ

The following table details the evolution of the relevant initials from Middle Chinese (MC) for both dialects corresponding to the above chain shifts.

    Taishanese   Cantonese  
    台山话   广府粤语  
1.L1 ts tʃ/tɕ ts
1.L2 t ts ts
1.L3 ? t t
           
2.L1 tsh h/tɕh tsh
2.L2 th tsh tsh
2.L3 h th th
           
3.L2 s ʃ s
3.L3 ɬ s s

Let's have a look at Chain Shift 1.  Cantonese [ts] is a merger of MC [tʃ], [tɕ], and [ts]. However, Taishanese [ts] contains only MC [tʃ] and [tɕ] while MC [ts] turned into [t]. Thus Cantonese [ts] initial corresponds to Taishanese initials [ts] and [t]. This accounts for the different in row 16 of the Cantonese/Taishanese Initial Mapping Table. The following stage of this chain shift, 1.L3, accounts for row 5.

A similar analysis of Chain Shift 2, the aspirated version, accounts for the differences of rows 6 and 17.

Finally, the fricative shift, Chain Shift 3, accounts for row 18.

Since the shifts mainly involve mainly affricate-fricative initials and the chains are long, I call them collectively the Great Taishanese Affricate-Fricative Consonant Initial Chain Shift (台山话塞擦音声母大链变).

Curiously, Taishanese syllables with mismatched initials under discussion in this blog (i.e. rows 5,6,16,17, and 18) belong to the MC chitou group (齿头音) which contains exactly five consonants [ts, tsh, dz, s, z]. That is, syllables start with these five consonants in the Middle-Chinese period today have different initial consonants between Cantonese and Taishanese, and their differences can be explained by the Chain Shift model discussed above. Is this just a coincidence?

This model covers all the asterisked rows of the Mapping Table except row 19.