Why are "binary $1+1$" and "Boolean $1+1$" not the same?

Question

Accepted Answer

Binary $1+1 = 10$ (with carry) but Boolean $1+1 = 1$

Answer

Both give 1

Answer

Binary $1+1 = 1$ but Boolean $1+1 = 10$

Answer

Both give 10

Why are "binary 1+11+11+1" and "Boolean 1+11+11+1" not the same?