From Laura K. Lawless,
Your Guide to French Language.

Think accents don't matter when writing in French? Think again!

You may not realize it, but French accents have a purpose. While some accents just signify that an S used to follow that vowel in Old French (e.g., étudiant used to be spelled estudiant), most French accents indicate the correct pronunciation of the letter they modify. In addition, there are dozens of French word pairs which are spelled (though not always pronounced) the same other than accents. To avoid confusion, you should always distinguish between these words by using the correct accents.*
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a - third person singular of avoir (to have)
à - (preposition) to, at, in

acre - acre
âcre - (adjective) acrid, pungent

âge - age
âgé - (adjective) old

aie - first person singular subjunctive and second person singular imperative of avoir
aïe - (interjection) ouch

arriéré - (adjective) overdue, backward; (noun) backlog, arrears
arrière - back, stern, rear, aft

bronze - bronze object
bronzé - past participle of bronzer (to tan, bronze)

ça - (indefinite demonstrative pronoun) that, it
çà et là - here and there

colle - glue
collé - past participle of coller (to glue)

congres - eels
congrès - conference, congress

cote - quotation, quoted value, rating
coté - highly thought of / rated (past participle of côter)
côte - rib, slope, coastline
côté - side

cure - cure, treatment
curé - priest; past participle of curer (to clean out)

crêpe - crepe (thin pancake), crepe paper
crêpé - past participle of crêper (to backcomb, crimp)

de - (preposition) of, from
dé - thimble, die

des - (indefinite article, partitive article) some; contraction of de + les
dès - (preposition) from

différent - different
diffèrent - third person plural conjugation of différer (to differ)

du - contraction of de + le
dû - past participle of devoir (to have to)

-e vs é
At the end of -er verbs, the accent is the difference between the first and third person singular present tense and the past participle
-e - étudie, parle, visite
-é - étudié, parlé, visité

entre - (preposition) between
entré - past participle of entrer (to enter)

es - second person singular of être (to be)
ès - contraction of en + les

êtes - second person plural of être
étés - summers

eut - third person singular passé simple of avoir
eût - third person singular imperfect subjunctive of avoir

ferme - farm
fermé - past participle of fermer (to close)

fut - third person singular passé simple of être
fût - third person singular imperfect subjunctive of être

gène - gene
gêne - trouble, bother, embarrassment
gêné - (adjective) short of, embarrassed; past participle of gêner (to bother)

grade - rank, degree
gradé - officer

haler - to haul in
hâler - to tan

illustre - illustrious, renowned
illustré - illustrated

infecte - (fem adjective) revolting, filthy, obnoxious
infecté - infected, contaminated

interne (adj) internal, inner; (noun) boarder, intern
interné - inmate (of a mental hospital), internee (politics)

jeune - (adjective) young
jeûne - fasting

juge - judge
jugé - past participle of juger (to judge)