As verbs, "wear" refers to the action of having clothes already on. For example:
"My friend is wearing a hat."
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"My friend currently has a hat on his head."
"Dress" means to put clothes on something. You can dress yourself or someone/something else. For example:
"Little girls love dressing up their dolls."
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"Little girls love putting clothes on their dolls."
"I have trouble dressing myself when I'm drunk."
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"I have trouble putting clothes on (myself) when I'm drunk."
We also like to replace "dressing oneself", which is formal, with the more natural "getting dressed".
"I need to get dressed because I have school in 30 minutes."
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"I need to put clothes on (myself) because I have school in 30 minutes."