Péřový míček a plastový míček jsou různé.

They are in the nominative case, because they are the subject of the sentence.

• péřový míček = nominative singular
• plastový míček = nominative singular

The sentence has two subjects joined by a = and.

Because the subject is plural in meaning.

Even though each noun phrase is singular by itself, they are joined by a:

• péřový míček
• a
• plastový míček

Together they form a plural subject, so Czech uses jsou = are, not je = is.

Because různé has to agree with the subject, and the subject is not masculine animate.

The noun míček is masculine inanimate. In the plural, predicate adjectives usually have:

• -í for masculine animate
• -é for masculine inanimate, feminine, and neuter

So:

• muži jsou různí = the men are different
• míčky jsou různé = the balls/shuttlecocks are different

That is why různé is correct here.

It is repeated for clarity.

Péřový míček a plastový míček clearly means two separate items. If you tried to avoid repeating míček, the sentence could sound less clear or more awkward.

Repeating the noun is very natural here because the speaker is contrasting one feather one with one plastic one.

Míček is a diminutive of míč.

• míč = ball
• míček = little ball, small ball

In Czech, diminutives are very common and often sound completely normal, not especially cute. So míček may simply be the usual word for a small sports ball or similar object.

They are adjectives made from nouns.

• plast → plastový = plastic
• peří → péřový = feather, feather-made, down-filled, depending on context

The ending -ový is a very common way to form adjectives meaning made of, connected with, or characterized by something.

A small extra note: péřový comes from peří, but the stem changes to péř-, so it is not formed in a completely transparent way for beginners.

The hardest part is ř, which is a special Czech sound.

A rough learner-friendly pronunciation guide for the whole sentence is:

PAY-rzhoh-vee MEE-chek a PLAS-toh-vee MEE-chek zhow ROOZ-neh

A few useful points:

• é, í, ý are long vowels
• č sounds like ch in check
• ř is a famous Czech sound, something like an r and zh sound pronounced together

You do not need to master ř immediately. Czech speakers are used to learners finding it difficult.

Sometimes, but různé fits this sentence better.

• různé = different, distinct from each other
• jiné = other, different

In this sentence, the idea is that the two shuttlecocks/balls are not the same, so různé sounds very natural.

Jiné is possible in some contexts, but it often feels a bit less direct here.

The given word order is the neutral one:

Péřový míček a plastový míček jsou různé.

Czech word order is more flexible than English, but different orders can sound more marked, emphatic, or stylistically unusual. For a learner, the version you have here is the safest and most natural basic pattern:

subject + verb + complement.

Because Czech agreement follows the whole subject, not each part separately.

You have:

• one péřový míček
• one plastový míček

Together, that makes two things, so the verb and predicate adjective are plural:

• jsou = are
• různé = different, plural form

So singular nouns can still produce plural agreement when they are joined by a.