Close, shut are very close synonyms in the sense of to stop or fill in an opening by means of a closure (as a door, a gate, a lid, or a cover) and are often used interchangeably.
However, they may have distinctive nuances of meaning and quite different implications in idiomatic use.
Close is the more general term, usually implying both the act of stopping an opening and the result produced by such an act but stressing exclusion of those who would enter or pass through.
Shut stresses the act or process and the means employed in this process; it not only carries a more emphatic implication or a more vivid suggestion of drawing a door, gate, lid, or window into a position which closes the opening, but it often also evokes an image of fastening securely (as by drawing a bar or a bolt or locking); hence, in closing a door or gate one merely draws it into a position which bars entrance or egress until it is again opened but in shutting a door or gate one pushes or pulls it into the position where it is closed.
Idiomatically, one closes (not shuts) an opening or a gap or one closes (not shuts) a park or a church to the public, because in neither case is the use of a door, gate, or other means of exclusion clearly or definitely implied.
On the other hand, in idiomatic use shut, especially when followed by up, out, or against, carries a stronger and often a more direct and emphatic suggestion than close of the interposition of a barrier or obstacle (often an immaterial one) that effectually prevents ingress or egress.