There is the issue of intimacy being "meshed" with tumah today, as mentioned in a different answer. We don't want to broadcast the fact of going to the mikveh.
There are a couple of other factors.
One, back in the day we wanted people to know that a woman was tameah. it was important that everyone be aware of her status so they don't accidentally become tameah or make any item tameah.
There's even a teshuvas hageonim discussing women having entirely different sets of household utensils- chairs, dishes etc. for when they are tameah. (It's referenced in Shu"t Binyan Tzion, and it comes up when discussing the Raavad's shita in Y.D. 193 regarding dam besulim. In the Shita mekubetzes hachadash, in the footnotes in the raavad on that topic, they quote the teshuva.)
The point is, being tameah was well known and was relatively difficult to keep hidden even if she tried. This is why we had the issue of סרך בתה the daughters learning from their mother's immersion as discussed in Y.D. 195.
One additional aspect of the tumas nida is mentioned in the famous Rabbeinu Bechaye regarding Lavan and Rachel in sefer Bereishis 31:35. When Rachel says that she can't stand up for Lavan because "the way of women is upon her" (presumably referring to nidus) Lavan doesn't say anything but leaves the tent.
Rabbeinu Bechaye explains that even the non-jews stayed away from nidos at that time and wouldn't even talk to them because of how powerful the tumas nida was.
Besides the tumah aspect, there's the practical aspect. Before modern female hygienic products were available, menstruation was a messy item.
In Shulchan Aruch Even HaEzer 73:1, when it describes the obligation for a husband to provide clothing for his wife, the be'er hagolah explains based on the gemara that it means he gives her new clothing during the rainy season and the old worn out clothing remain hers to use while she's a nida.
Practically women didn't want to wear nice clothing which would get stained and ruined. So they kept their shmattes to wear monthly.
Seemingly this applied even after the time of the Shulchan aruch.
Today, things are changed on all three fronts.
1) Tevilah is now almost exclusively done for intimacy.
2) We are not concerned about women making things tameah
3) Especially with feminine hygiene products, women are capable of hiding their nidus entirely.
Thus, we have reason to hide it (tznius), no reason not to hide it (tumah's not an issue) and the option of hiding it (hygiene).
[edit: these points were what came out while learning hilchos nida in kollel in Yerushalayim and discussing these topics with various morei Horaah. I don't remember which rabbonim discussed which points, but these points were all commonly assumed to be the reason for the switch. The last point is especially true- it only really became relevant to discuss hiding menstruation once hygienic products made it possible to hide. That eliminated the practical need for "bigdei nidus."]
It's worth noting that Rav Moshe Feinstein famously discusses this in his teshuva(Igros Moshe Y.D. 2:77) about a couple publicly carrying a baby carriage on a bus while she's a nida.
The questioner wanted to be lenient, even though during nidus the couple is not supposed to carry things together (per Y.D. 195) because of kavod habriyus. If everyone sees them not carrying, it will be obvious to the bystanders that she's a nida and she'll be embarrassed.
Rav Moshe responds that there's nothing to be embarrassed about having one's period. He brings proof from the fact that in the olden days women wore bigdei nidus to publicize the fact that they were tameah. He uses that as a proof that even nowadays there's no problem of publicizing her tumah (this is not the same as the need to hide one's tevilah, as he discusses.)
For more about the teshuva from Rav Moshe see this answer