Barattolo vs. vasetto
Both the words barattolo and vasetto translate to the English word jar, but they differ significantly in terms of size, common materials, and the types of products they typically hold.
Barattolo
A barattolo refers to a medium to large, usually cylindrical container that can be made of glass, metal (like tin), or plastic. It is typically used for storing dry pantry goods like coffee and sugar, canned foods like peeled tomatoes and beans, or non-food items such as paint.
Ho comprato un barattolo di vernice blu per dipingere la stanza.
(I bought a jar of blue paint to paint the room.)
Metti la farina in quel barattolo di vetro grande.
(Put the flour in that large glass jar.)
Mi serve un barattolo di pomodori pelati per fare il sugo.
(I need a can of peeled tomatoes to make the sauce.)
Il barattolo del caffè è quasi vuoto in cucina.
(The coffee jar is almost empty in the kitchen.)
Ha nascosto i biscotti al cioccolato in un barattolo di latta.
(He hid the chocolate cookies in a tin jar.)
Vasetto
A vasetto is a diminutive form of the word vaso and refers to a small jar. It is almost exclusively made of glass, though small plastic containers like those for yogurt are also included. It is normally used for preserves, jams, honey, pesto, baby food, or small cosmetic face creams.
Ogni mattina a colazione mangio un vasetto di yogurt bianco.
(Every morning at breakfast I eat a small jar of plain yogurt.)
La nonna mi ha regalato un vasetto di marmellata alle fragole fatta in casa.
(Grandma gave me a small jar of homemade strawberry jam.)
Ho quasi finito il mio nuovo vasetto di crema per il viso.
(I have almost finished my new small jar of face cream.)
Per questa ricetta serve un vasetto di pesto alla genovese.
(For this recipe you need a small jar of Genoese pesto.)
Il bambino ha appena mangiato un vasetto di omogeneizzato alla mela.
(The baby just ate a small jar of apple baby food.)
Summary
In summary, use barattolo when referring to larger, bulkier containers made of glass, metal, or plastic that hold items like coffee, canned vegetables, or paint. Choose vasetto for small, mostly glass jars that contain sweet preserves, yogurt, sauces like pesto, or cosmetics.