Quick answer
Ш is a hard sh, tongue pulled back. Щ is a long, soft sh, tongue against the roof of the mouth — like “fresh sheets” said quickly.
Щщ
Щщ
shch
шко́лаschool — Ш
How to say it
For Ш, say “shhh” as if hushing someone. For Щ, smile and let the sound go further forward and longer.
шко́лаschool — Ш
ша́пкаhat — Ш
борщborscht — Щ
ещёstill, more — Щ
това́рищcomrade, mate — Щ
же́нщинаwoman — Щ
The borsch problem
The soup is борщ with Щ, which is why English spells it “borscht” or “borshch” and nobody agrees. Say it soft and long and you are closer than either spelling.
A pair that proves it
Чашка versus ча́ща, or шит (sews) versus щит (shield). Same consonant family, different words.
Typical mistakes
Pronouncing Щ as “shch” with a hard ch
one long soft sh
Modern standard Russian lost the ch part — it is simply a long, soft sh.
борш
борщ
The word ends in Щ, and the soft ending is audible.
Check yourself
Which word contains Щ?
шко́ла — school — Ш.