～て欲(ほ)しいI want you to

Both ように～てほしい and てほしい express wanting someone else to do something, with the key difference being the presence of よう. よう adds a nuance of doing something in a certain way. For example, 忘れないでほしい ('I want you not to forget.') vs. 忘れないようにしてほしい ('I want you to make sure to not forget.', in the sense of doing what needs to be done to ensure you do not forget).

てほしい and がる both relate to expressing desire but are used differently. てほしい is used when you want someone to do something, e.g. これを買ってほしい ('I want you to buy this.'). がる describes what another person appears to feel, e.g. 田中さんは車を買いたがっている ('Mr. Tanaka seems to want to buy a car.').

てほしい and たがる both express desire but in different ways. てほしい is used when the speaker wants someone else to do something, e.g. これを買ってほしい ('I want you to buy this.'). たがる combines the たい-form with がる to describe a third person's apparent desire, e.g. 田中さんは車を買いたがっている ('Tanaka seems to want to buy a car.').

Both がほしい and てほしい express a want, but differ in meaning and usage. がほしい follows nouns to express the speaker's desire for something, whether physical, e.g. 金がほしい ('I want money.') or figurative, e.g. 自由がほしい ('I want freedom.'). In contrast, てほしい uses the て form of a verb to express a desire for someone else to do something, e.g. これを見てほしい ('I want you to look at this.').

Both たい and てほしい express a desire, with one key difference. The たい form is used to express your own desires, similar to the English 'I want to...' while てほしい expresses your desires for other people, or in other words, 'I want you to…' For example, 見たい would express your desire to see something, while 見てほしい would be used if you want someone else to take a look.