For me, "get dressed" and "put clothes on" imply you are starting out naked, or perhaps wearing only nightwear and getting ready for the day ahead.
Whereas "dress" on it's own seems to be most often used when someone is already wearing clothes and needs to change, usually for a specific purpose, and implies the change of clothing is to something more suitable. For example:
"It's 7 o'clock, we need to dress for dinner" - we need to take off these casual day clothes and put on more formal items before our guests arrive for a fancy meal. "...get dressed..." here would sound like you had a lazy day and had not yet got out of your pyjamas.
"You should dress warmly, they're forecasting snow" - change what you are wearing or add more layers in preparation for going outside.
"They dressed for the beach and grabbed their towels" - maybe they put on swimwear under their clothes, or simply put on lighter clothing for a day in the sun. Again, "got dressed" would sound like they had not yet done so today, whereas "dressed" alone sounds like getting changed, and dressing [in a suitable way for] the beach
With adverbs (and especially in the past tense), "dressed" does not sound so stilted, and seems OK as a replacement for "got dressed":
"He dressed quickly before her husband came up the stairs"
When doing it to someone / something else, "dress" seems better (than "put clothes on" - "got dressed" would be wrong):
"The shopkeeper dressed the mannequin in the new summer range of sportswear"
"She dressed her daughter and they went to the playground"
Also in the same vein, you dress a salad, a window, or other things that might be decorated - a table, a turkey, or a Christmas tree perhaps.
Consider when emphasis is needed regarding who is doing the dressing, such as whether an elderly person is capable of putting clothes on without help:
"Luckily, she can still dress herself" sounds fine, and simpler than "Luckily she can still get dressed by herself" (which is also fine but longer)