Precedence only affects parsing. The same op is used in the resulting tree for both of these operators.

>perl -MO=Concise -e"$a and $b"
6  <@> leave[1 ref] vKP/REFC ->(end)
1     <0> enter ->2
2     <;> nextstate(main 1 -e:1) v ->3
-     <1> null vK/1 ->6
4        <|> and(other->5) vK/1 ->6
-           <1> ex-rv2sv sK/1 ->4
3              <#> gvsv[*a] s ->4
-           <1> ex-rv2sv vK/1 ->-
5              <#> gvsv[*b] s ->6
-e syntax OK

>perl -MO=Concise -e"$a && $b"
6  <@> leave[1 ref] vKP/REFC ->(end)
1     <0> enter ->2
2     <;> nextstate(main 1 -e:1) v ->3
-     <1> null vK/1 ->6
4        <|> and(other->5) vK/1 ->6
-           <1> ex-rv2sv sK/1 ->4
3              <#> gvsv[*a] s ->4
-           <1> ex-rv2sv vK/1 ->-
5              <#> gvsv[*b] s ->6
-e syntax OK
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And check out "if":

>perl -MO=Concise -e"if ($a) { $b }"
6  <@> leave[1 ref] vKP/REFC ->(end)
1     <0> enter ->2
2     <;> nextstate(main 3 -e:1) v ->3
-     <1> null vK/1 ->6
4        <|> and(other->5) vK/1 ->6
-           <1> ex-rv2sv sK/1 ->4
3              <#> gvsv[*a] s ->4
-           <@> scope vK ->-             \ The only difference
-              <0> ex-nextstate v ->5    / is the extra scope.
-              <1> ex-rv2sv vK/1 ->-
5                 <#> gvsv[*b] s ->6
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