If your question seeks an analysis of the whole of Hamletn, this answer format does not permit for such a large task. I can give you a brief look at the comparison. Quarto editions of Shakespeare's works are recognized by experts as unauthorized versions of his plays derived from playbooks (scripts) or other more doubtful sources. Folios, on the other had were professionally prepared editions that show no evidence of having been authorized by Shakespeare and that vary, sometimes radically from the Quarto editions. Quarto 1 is the original publish version of Hamlet, 1603. Quarto 2 followed the next year in 1604. Folio 1 came twenty years after the first, in 1623.
To work backwards, Folio 1 is derived from Q2 and, in the main, identical or very nearly similar. For example, the opening of Act I, scene i, illustrates this as it is identical in both:
Bar. VVHose there?
5Fran. Nay answere me. Stand and vnfolde your selfe.
Bar. Long liue the King,
10Fran. You come most carefully vpon your houre,
Bar. Tis now strooke twelfe, get thee to bed Francisco,
Fran. For this reliefe much thanks, tis bitter cold,
And I am sick at hart.
Bar. Haue you had quiet guard?
15Fran. Not a mouse stirring.
Bar. Well, good night:
WHo's there ?
Fran. Nay answer me: Stand & vnfold
Bar. Long liue the King.
Fran. You come most carefully vpon your houre.
Bar. 'Tis now strook twelue, get thee to bed Francisco.
Fran. For this releefe much thankes: 'Tis bitter cold,
And I am sicke at heart.
Barn. Haue you had quiet Guard?
Fran. Not a Mouse stirring.
Barn. Well, goodnight.
Compare this to the original Quarto 1 and we see a surprising difference:
1. STand: who is that?
2. Tis I.
101. O you come most carefully vpon your watch,
2. And if you meete Marcellus and Horatio,
The partners of my watch, bid them make haste.
1. I will: See who goes there.
Enter Horatio and Marcellus.
You can identify the words that are different, for instance, Q1 "Stand: Who is that?" is replaced by Q2 "Whose there?" and the correct F1 "Who's there?" The effect of changes early in I.i is to prolong the discussion of the appearance of the ghost and to more explicitly establish the soldiers' loyalty to the fallen King of Denmark.
More complex changes are found, as an example, in Hamlet's famous III.i "To be" speech. For one thing, the Q1 speech is about one-half as long as Q2 and F1. The effect of these changes is to alter the emphasis of the speech from Q1 compared to the others. Q1 is a contemplation of the uncertainties of death--which leads to the presence of the Judge, "an euerlasting Iudge"--in the face of evils and injustices of life:
Scorned by the right rich, the rich curssed of the poore?
The widow being oppressed, the orphan wrong'd,
The taste of hunger, or a tirants raigne,
And thousand more calamities besides,
To grunt and sweate vnder this weary life, (Q1, 1603)
This rather straightforward emphasis is changed in Q2 and continued in Folio 1 to that of a philosophical musing on the nobility of suffering:
Whether tis nobler in the minde to suffer
The slings and arrowes of outragious fortune,
Or to take Armes against a sea of troubles,
These changes between the original Quarto and the later Q2 and F1 manuscripts does much to alter the characterization of Hamlet and make his motivation richer as well as more complex and, one might say, more confusingly elusive.