Mi abuela sabe coser un botón, pero yo solo sé doblar la ropa.

Because saber + infinitive means to know how to do something.

• Mi abuela sabe coser = My grandmother knows how to sew
• Conocer is used for being familiar with people, places, or things:
  • Conozco Madrid
  • Conoce a mi abuela

So in this sentence, saber is the correct verb because it is talking about a skill.

After saber, Spanish normally uses the infinitive to express a learned ability or skill.

• sabe coser = knows how to sew
• sé doblar = I know how to fold

This is very similar to English know how to + verb. The verb after saber does not get conjugated here because sabe and sé are already the conjugated forms.

The accent distinguishes sé from se.

• sé = I know from saber
• se = a pronoun, as in se lava

So in yo solo sé doblar la ropa, the accent tells you this is the verb form I know.

A useful extra note: sé can also be the affirmative command of ser:

• Sé amable = Be nice

But in this sentence, it clearly means I know.

Yes, it could. Spanish often drops subject pronouns because the verb ending already shows who the subject is.

• solo sé doblar la ropa is perfectly correct

Here, yo is included for contrast:

• My grandmother knows how to sew a button, but I only know how to fold clothes.

So yo adds emphasis, especially because the sentence is comparing two people.

Here solo means only.

• yo solo sé doblar la ropa = I only know how to fold clothes

In modern standard Spanish, this is usually written solo without an accent, even when it means only.

Older spelling, or some writers for clarity, may use sólo, but current standard usage generally prefers solo unless there is real ambiguity. In this sentence, solo without an accent is completely normal.

They are used differently because they refer to different kinds of things.

• un botón = a button / one button
  It refers to a single, unspecified button.
• la ropa = clothes / the clothes
  Ropa is a collective noun, and Spanish often uses the definite article for things spoken of in a general or natural way.

So:

• coser un botón = sew on a button
• doblar la ropa = fold the clothes / fold clothes

Even when English might leave out the, Spanish often includes the article.

Because in Spanish, coser un botón already naturally means to sew on a button.

The button is the direct object, so Spanish does not need a separate word equivalent to English on in this basic expression.

If you want to say where the button is being sewn, you can add that:

• coser un botón a la camisa
• coser un botón en la camisa

But by itself, coser un botón is a normal way to say sew on a button.

You could say it, but the meaning changes slightly.

• saber coser un botón = to know how to sew a button, to have the skill
• poder coser un botón = to be able to sew a button, to be capable in that moment or circumstance

So:

• Mi abuela sabe coser un botón focuses on her know-how or skill.
• Mi abuela puede coser un botón could mean she is able to do it, perhaps because she has time, materials, or physical ability.

In this sentence, saber is better because the contrast is about what each person knows how to do.

Because the first part of the sentence is not negative.

• Mi abuela sabe coser un botón, pero yo...
  This is a simple contrast, so pero is correct.

Sino is typically used after a negative statement to correct or replace something:

• No sabe coser, sino tejer

So:

• pero = but
• sino = but rather / rather

This sentence needs pero.

Yes: coser is often confused with cocer.

• coser = to sew
• cocer = to cook / boil

They are different verbs with different meanings, and they may sound different or the same depending on accent.

In much of Spain:

• coser has an s sound
• cocer has a th sound

So for learners of Spain Spanish, that distinction is especially useful.

Because Spanish normally does not use the article before a family noun when it is preceded by a possessive.

• mi abuela
• tu hermano
• nuestra madre

So la mi abuela would be wrong in standard modern Spanish.

This is the same pattern as:

• mi casa
• mi libro

The possessive already makes the noun definite, so no article is needed.