First, we evaluate the first pair of parentheses: \begin{align*} (t) \Join (t+2) &= (t + 2)((t+2) - 3) \\ &= (t+2)(t - 1)\\ &= t^2 + 2t - t - 2\\ &= t^2 + t - 2.\end{align*}Next, we evaluate the second pair of parentheses: \begin{align*} (t + 1) \Join (t+1) &= ((t+1) + 2)((t+1) - 3) \\ &= (t+3)(t - 2) \\ &= t^2 + 3t - 2t - 6 \\ &= t^2 + t - 6.\end{align*}Subtracting the two expressions and noting that certain terms cancel, we obtain $ (t^2 + t - 2) - (t^2 + t - 6) = -2 - (-6) = \boxed{4} $.