Another (python) option, offering a bit of a "fancy" report:
----------------------------------------------------------
Newly copied from /home/jacob/map 1 to /home/jacob/map 2:
- Naamloos document
- pscript_2.py
- test123
- monkey_out.txt
----------------------------------------------------------
Unique files in /home/jacob/map 2:
- file_in_targetfolder
----------------------------------------------------------
It does what you describe:
- it copies files from folder A to B if they do not (yet) exist in B
- reports files in B, which are not in A
The script
#!/usr/bin/env python3
import sys
import os
import shutil
folder_1 = sys.argv[1]; folder_2 = sys.argv[2]
files_1 = os.listdir(folder_1); files_2 = os.listdir(folder_2)
# make the length (n) of separating line "smart"
s = "\nNewly copied from "+folder_1+" to "+folder_2+':'; n = len(s)
print("\n"+n*"-"+s)
for f in files_1:
f1 = folder_1+"/"+f
if all([os.path.isfile(f1), not f in files_2]):
shutil.copyfile(f1, folder_2+"/"+f)
print("-",f)
print(n*"-"+"\nUnique files in "+folder_2+":")
for f in files_2:
f2 = folder_2+"/"+f
if all([os.path.isfile(f2), not f in files_1]):
print("-",f)
print(n*"-")
How to set up
Copy the script into an empty file, save it as sync_report.py
Test- run it by the command:
python3 /path/to/sync_report.py <folder_a> <folder_b>
If the directories contain spaces, put them between single quotes.
If all works fine, use it as above, or:
- make the script executable
add the following command to a shortcut key:
gnome-terminal -e "'/path/to/sync_report.py' '/path/to/folder_a' '/path/to/folder_b'"
Choose: System Settings > "Keyboard" > "Shortcuts" > "Custom Shortcuts". Click the "+" and add the command.
Then, when the key combination is pressed:

Don't let the background distract you, my terminal is semi-transparent. :)
Alternatively, as mentioned by @ParanoidPanda (thanks for mentioning), add it to your ~/.bashrc file. Create an alias which calls the script:
alias <nameofalias>='python3 /path/to/sync_report.py <folder_a> <folder_b>'