There seems to be two aspects:
One is to get the end points to coincide. You may tick View>Snap Guides, then click one path by the end Anchor Point and drag it to snap (Smart Guides say anchor when you are there).
The other is to get rid of the gap in the stroke(s), You may join the paths into one, dragging across the coinciding Anchor Points with the Direct Selection Tool and Ctrl/Cmd+J, or you may keep the separate paths and change the Cap to Round in the Stroke palette/panel which will give you a rounded outer appearance); I presume the former is what you wish, in which case you may choose between Round and Miter Join.
You are welcome, Brigid.
Black pointer: Select the two paths.
Object>Path>Join (Ctrl J)
If there are two anchors where you want just one: Delete Anchor Point Tool (the minus Pen): Click one of the Anchor Points.