How do the men and women differ in their separate investigations of Mr. Wright's murder? Why are their approaches so dissimilar?

Susan Glaspell's play Trifles is primarily concerned with the way men devalue and dismiss the wants, needs, thoughts and emotions of women. The play is centered around a married couple, John and Minnie Wright, neither of whom ever appears on stage. Instead, the men and women who make up the cast of the play can be understood as proxies for The Man (John Wright) and The Woman (Minnie Wright) at the heart of the narrative.
The men are Sheriff Peters, Mr. Hale, and the county attorney Mr. Henderson. The women are Mrs. Peters and Mrs. Hale. Before the play has even begun, the men outnumber the women, and two of them—the sheriff and the attorney—are in positions of authority. The women are merely wives without any social authority; they are adjuncts to their husbands and have come along today to do the "women's work" of collecting a few items to take to Minnie Wright in jail. The men are here for the ostensibly higher purpose of determining how John Wright was murdered and, more particularly, why.
The men investigate the Wright property for clues to the murder, while their wives sit in the kitchen and talk about Minnie Wright and what might have happened. This set-up reflects the traditional social understanding of men as "active" versus women as "passive" actors in events. The men are deliberately seeking out answers to specific questions. The Sheriff and Mr. Henderson quiz Mr. Hale to get the bare facts of the matter. When Mr. Hale remarks that he "didn't know as what [John's] wife wanted made much difference to John," he has inadvertently hit upon the reason for the murder. This is not evident to Mr. Henderson, however, who dismisses this observation and says,

Let's talk about that later, Mr. Hale. I do want to talk about that, but tell now just what happened when you got to the house.

The men open and shut drawers, rifle through the cupboards, pick things up and put them down, go upstairs to the scene of the murder and outside to the barn, all on their quest for answers. Mrs Hale, like her husband, also suggests Minnie was unhappy in her marriage, saying, "I don't think a place'd be any cheerfuller for John Wright's being in it." Mr. Henderson once again dismisses the idea that Minnie's feelings had any real bearing on the murder:

I'd like to talk more of that a little later. I want to get the lay of things upstairs now.

The men troop upstairs to investigate the crime scene, leaving the wives to themselves. The women have already noted that it's hard work being a farmer's wife and that men don't make it any easier, but the men just laugh at them in gentle derision. In one of the cupboards, Minnie Wright's jars of fruit preserves have frozen and exploded, and Mrs. Peters says, "She was worried about that when it turned so cold. She said the fire'd go out and her jars would break." The Sheriff thinks this is ridiculous, and exclaims

Well, can you beat the woman! Held for murder and worryin' about her preserves.

Mr. Henderson agrees:

I guess before we're through she may have something more serious than preserves to worry about.

And Mr. Hale chimes in:

Well, women are used to worrying over trifles.

The men have a very set idea of what is "important" and what is not. Anything women care about is, by default, unimportant, a "trifle."
The women, by contrast, see things very differently. Theirs is a life in full view of and yet strangely invisible to the men. As they move around the kitchen, waiting for the men to finish investigating, they notice a great deal of meaning in the "trifles" the men so readily dismissed. The shabbiness of the place, to the men, meant Minnie was lazy and slovenly; but to the women, it means she was lonely and sad. Mrs. Hale remembers her when she was Minnie Foster, a "lively" girl who "used to wear pretty things" and sing "in the town choir." She and Mrs. Peters discuss how living in this isolated place with someone like John Wright—someone cold, distant, and unkind—must have worn Minnie down and changed her personality over the past thirty years.
As the women putter about, Mrs. Hale picks up a piece of Minnie's quilting from the work-basket. Most of the piece shows how "nice and even" Minnie's stitching normally was, and then, suddenly, "It's all over the place! Why, it looks as if she didn't know what she was about!" The men haven't bothered to look at the quilting, deeming it of no importance, but the details Mrs. Hale sees in the stitching reveal that Minnie had some kind of emotional crisis.
As Mrs. Peters looks for paper and string to wrap up her little package of items to take to Minnie, she finds a bird cage in the cupboard—the same cupboard the men rummaged in earlier, without noticing the bird cage at all. She pulls it out of the cupboard and she and Mrs. Hale speculate about the fate of the bird. There is no cat about the place, for as Mrs. Peters says, Minnie is afraid of cats, so a cat couldn't have eaten the bird. The door of the bird cage is hanging off its hinges and "looks as if someone must have been rough with it."
This find makes the women oddly uneasy, and Mrs. Peters says,

I awful glad you came with me, Mrs. Hale. It would be lonesome for me sitting here alone.

Mrs. Hale agrees it would be lonesome, and expresses regret that she never bothered to visit Minnie:

I could've come. I stayed away because it weren't cheerful-and that's why. I ought to have come. I - I've never liked this place [...] I dunno what it is, but it's a lonesome place and always was. I wish I had come over to see Minnie Foster sometimes.

She says John Wright was "a hard man," and just speaking to him was like standing in "a raw wind that gets to the bone." She understands why Minnie would have wanted a bird and wonders again what happened to it. Feeling sorry for Minnie, she suggest Mrs. Peters take the quilt from the work-basket for Minnie to work on in jail, to "take up her mind." When Mrs. Peters lifts the quilt, she finds a small box, inside which is the body of the missing bird. Its neck has been broken. The women know John Wright must have done it, and Minnie's crime now makes complete sense—strangling John in his sleep to punish him for strangling her little bird.
Throughout the women's conversation, the men walk in and out of the kitchen a few times, talking to each other, completely uninterested in what their wives are discussing. "They've got awful important things on their minds," as Mrs. Peters says to Mrs. Hale—things so important that the men fail to notice the evidence all around them, looking instead for something obvious to them, something over which could be what they deem worth killing. The women, by contrast, in their "worrying over trifles," have determined the entire course of events, not just of the murder, but of Minnie's miserable married life. The house is run-down, the husband was cruel and controlling, the wife was lonely and isolated, and the one thing which might have brought her some companionship, some joy, was ruined by the husband, purely out of his spite. The "trifles" are the motive for the murder; more than that, they are its justification. The men do not see these things because the men do not see the women, except as minor players in a boring story happening outside of the main event: themselves. This is the attitude that oppressed Minnie Wright for thirty years.
Mrs. Hale and Mrs. Peters take revenge for Minnie by taking advantage of that attitude. Knowing the men do not care what they've been up to, they hide the bird's body, and with it, the motive. Without a clear motive for murder, Minnie Wright will be acquitted. If she killed her husband over a "trifle," the absence of that same "trifle" will set her free.