"A proof is a complete explanation. Sometimes a partial explanation suffices." Reuben Hersh in 
Nonetheless, we need to be alert: if diagrams are not made with care and accuracy, they may cheat us (examples); furthermore, even accurate pictures and diagrams may have some properties that are not typical of the general case. Although diagrams may convince and inform us, they prove nothing and can not be accepted as a proof of the statement. Some picture or diagram may seem convincing simply because we are not able to find a counter-example.
One thing is certain: analysing proofs without words may help our students to practice the interpretation of diagrams and then to find by themselves the mathematical facts they suggest.