Bash: let statement vs assignment
let does exactly what (( )) do, it is for arithmetic expressions. There is almost no difference between let and (( )).
Your examples are invalid. var=${var}bar is going to add word bar to the var variable (which is a string operation), let var+=bar is not going to work, because it is not an arithmetic expression:
$ var='5'; let var+=bar; echo "$var"
5
Actually, it IS an arithmetic expression, if only variable bar was set, otherwise bar is treated as zero.
$ var='5'; bar=2; let var+=bar; echo "$var"
7