Please show me example sentences with what does ありがたい means and how is it different from ありがとう?.
Tell me as many daily expressions as possible.

See a translation

ありがたい
①［感謝する］be grateful，be thankful，appreciate O〈Oが〉
▼前もってお知らせいただければありがたいのですが　I would be grateful [thankful] if you would let me know in advance.
▼「その箱を運ぶのを手伝いましょうか」「そうしていただけるとありがたいです」　“Shall I help you carry the box?”“｢I'd appreciate that. [That would be most appreciated.]”
▼「明日，うちに夕食に来ない？」「ありがたいんだけど，あいにく先約があって」　“Why don't you come to my house for dinner tomorrow?”“I'd love to, but I haave a previous appointment.”
②［自分にとって都合がよい］
▼ありがたいことに店が開いていた　Fortunately [Luckily] the store was open.
▼円安ドル高は輸入業者にとってはありがたくない状況だ　A weak yen and a strong dollar aren't welcomed by importers.
▼このお金はとてもありがたかった　［＝役立った］ This money ｢was really useful （to me） [≪略式≫came in handy].
③［尊い・もったいない］
▼ありがたいお言葉をいただく　receive some kind words

｢ありがたい｣ is an i-adjective which describes the speaker's gratitude/appreciation.
I guess it comes from "It hardly/rarely happens. / It's hard to exist."(｢ある (exist, happen [verb]) + がたい (hardly, rarely)｣).

｢ありがとう｣ is a phrase, "Thank you.".