These are grafted trees, meaning the desired ornamental willow tree above is grown onto the trunk of a root stock tree--an ordinary green colored willow with better hardiness or disease tolerance. 

These green sprouts are probably coming from below the graft union where the two grew together.  Thus, they are coming from the root stock. Prune them off and prune off any more that sprout from low on the trunk. 

Ellen

So should we cut off the two large  branches coming from the main trunk shown in these pictures or just go in and cut the branches with the green leaves within the tree?

Cut them off at the trunk.  

If any more solid green-leaved ones sprout like them, cut them off at the trunk as soon as you see them. 

Ellen