When you point at the object, you say "вот [он/она/оно]". You can also specify, that it is "вот тут", "вот там", etc. Тут and здесь are basically the same, but "тут" is more informal and colloquial than "здесь". They mean that the object is here, near you.

"Здесь" is an adverb and it refers to the place in/at which something is located: Я живу здесь в Москве. Мы здесь на этом сайте для того, чтобы учить языки (The opposite of "здесь" is "там": "Я нахожусь здесь в этой комнате, она находится там в другой комнате"). "Вот" is a particle, and as is usually the case with particles can be used in a variety of contexts. Generally, you use it when you are pointing or nodding to something (to some object). Eg.: Какая из этих машин твоя? -- Вот эта. When you are pointing to the place "здесь" and "вот" can follow one another. Eg.: Где мне расписаться? (поставить подпись) -- Вот здесь (pointing to the line in the blank) .

Actually, there is no much difference, it's the matter of style. Probably, 'тут' is more colloquial. The mockney variation of it may be 'тута' (Ты где,​ Анюта? - А я здеся, я тута). 'Здесь' is more used with the meaning of place. 'Вот' is more for drawing your attention with the meaning of 'здесь'.

Thanks for the help!

You can say to someone "Вот мой дом" and here you can see it, you're standing near your house and you point on it and say this. Or in the shop "Вот туфли, которые мне нравятся". So, I think that вот is when you point on some object that you see at the moment (Even if you take something out like you've said, when you say вот the thing you're pointing should be infront of you) Тут and здесь are synonims and I think there will be only few situations (but I have now no examples for you now) where you can't use тут instead of здесь. Здесь/тут раньше был магазин Он стоял здесь/тут Ты видел мой телефон? Он был здесь/тут