Your question, which is an important one, is the difference between two women—queens—in terms of their power. One is Elizabeth I of England, and the second is the character of Gertrude in Shakespeare's Hamlet.
When Elizabeth I came to the throne, she was twenty-five. She reigned alone, without a husband, for forty-five years. She was extremely intelligent and had inherited her father's shrewd and, sometimes harsh, side. She loved her country and served it well, but she would not compromise the integrity of England for anyone: not someone she was very fond of, as with the Earl of Essex—who was executed for treason, or her cousin, Mary, Queen of Scots, also executed for treason.
Elizabeth I commanded respect not only from her people and her political "staff," but also from foreign powers. She stood up to the rulers of the other major powers of the time: France and Spain, and held her own. She was able to steer a conservative path between the Catholics and Protestants of the time, and refurbished England's treasury. Under her reign, the English defeated the Spanish Armada, believed to be the strongest naval fleet in the world at that time. She also ushered in the English (or Elizabethan) Renaissance which brought about a renewed interest (rebirth) in the arts. Elizabeth ruled on her own as Queen, and though she had advisors, she ruled as she saw fit. She was an extremely powerful monarch.
Gertrude, on the other hand, is a much weaker woman. She is Queen by virtue, first, of her marriage to her dead husband, Old Hamlet. Upon his death, and seeming much too quickly, she marries her brother-in-law less than two months later, according to Hamlet.
Gertrude's hasty marriage is unseemly with regard to how quickly she married; and Elizabethans also considered that to marry a dead spouse's brother or sister was considered incestuous, so Gertrude is seen in a bad light for this reason as well. Hamlet is very disappointed with his mother.
In his "Oh, that this too, too solid flesh would melt" soliloquy in Act One, scene two, Hamlet speaks of how quickly his mother has remarried.
A little month, or ere those shoes were old
With which she follow'd my poor father's body…
O God! a beast that wants discourse of reasonWould have mourn'd longer… (150-154)
Later, Hamlet shares with Horatio that the wedding so closely followed the funeral, that leftovers from the first could have been served at the second. (This is hyperbole.)
Thrift, thrift, Horatio! The meats baked for the funeral
Were also put on the marriage tables. (I.ii.185-186)
Gertrude seems to have married Claudius because she was weak—perhaps not a weak woman, per se, but one who wielded no power in her male-dominated society. There is nothing that indicates that Gertrude was drawn to Claudius before Old Hamlet's death. In fact, Hamlet reports that she worshipped the ground his father walked on. Claudius might have persuaded Gertrude to marry: this would have solidified his place upon the throne. (Hamlet has no concerns that he is not king.) Married to Claudius, she is cared for, and so, too, is Hamlet. Gertrude very much loves her son and worries for him; in this way, she is a good woman. She acts the part of the happy newlywed with Claudius—but is there a choice?
Gertrude as a woman who is queen in name only. And by the play's end, she is not only crushed by the news of Claudius' crime, but can do nothing—she cannot save herself or her son.
These are very different women.