Note: the Difference tool (as suggested by EricLohmann) should work. Since it didn't, here's another method to try.
Step 1: Use the Spatial Query tool to select overlapping points.
Find the Spatial Query tool in the Vector Menu. You may need to enable it from the Plugin menu > Manage and install plugins.
Use the Spatial Query tool to select source features from the first layer where the feature equals reference features of the second layer.
If all the identical features are selected, proceed to step 3. If not, it may be because the points are very close together but not identical.
Step 2: Buffer the second layer and repeat the spatial query using the buffer layer.
Vector menu > geoprocessing > fixed distance buffer
Choose a buffer distance small enough that the closest non-identical points will not fall within each others buffers. My closest non-identical points are 0.006 degrees apart, so I used a buffer of 0.001 degrees.
Repeat step 1, using the first points layer and the new buffer layer.
Step 3: Delete the selected points from the first point layer.
Turn on editing for the layer. Use the Delete selected button. Save edits.