Public Torah reading (Kri'at Hatorah Betzibbur) is always done from a parchment.
I have seen some congregations read the Haftarah from a parchment, as well. But, this seems to be a minority. It seems, then, that there must be a halachic requirement to read from parchment for the Torah but, there is no unanimous agreement that one must use a parchment for the Haftarah. Why this difference?
You aren't the first person to wonder about this. The Levush wrote "All my days I wondered why I never saw in any place the practice to write the Haftarot like a proper book as we do for Esther" (OC 284).
In truth though having a full set of Neviim is expensive and already in the time of the Gemara (Gittin 60a) they permitted writing out just the needed Haftarot (called a Sefer Aftarta) and not the whole books (even though in general it is prohibited to write incomplete potions of books of Tanakh) because עת לעשות לה׳ it is worth violating certain laws to avoid the practice being lost entirely.
The Magen Avraham (OC 284) writes that "nowadays" with the advent of the printing press (which he considered Kosher writing) complete books became cheaper so writing Sifrei Aftarta returned to being prohibited. He finds a minority opinion in Rishonim that doesn't require proper black ink (Dyo) for Neviim and extends that to allowing paper as well. Thus a printed book of Navi would be considered Kosher.
Today books are printed from computers (not hand operated printing presses) so it's highly questionable if the Magen Avraham's reasoning would still apply, let alone that it's relying on extensions of certain minority views. Plus it's also highly questionable if bound books are considered complete books (recall originally it was one long continuos scroll written on one side, not multiple sheets of paper written on both sides) which was the whole advantage of the Magen Avraham's position anyway.
Thus especially today it makes sense to prefer a properly written Sefer Aftarta over printed works as was the practice for centuries (Eliyah Rabba 284, Arukh haShulchan 284, Yechavveh Da'at 5:26, Nefesh HaRav pg 161, Shulchan Shlomo 284, Chazon Ish (Shabbat) 6:11, Chevel Nachalato 17:12).
If you have the money to buy a full set and not rely on the leniency in the Gemara, all the better!
If all you have is printed books, most are lenient to still do the Haftara, because the Gemara already allowed ignoring laws to allow the Haftara to be read. It should be stressed that עת לעשות לה׳ is literally the most Bedieved justification that can be given to a practice.
The Mishna Berura (OC 284 Shaar HaTziyun 4) writes:
דבעיקר הדין יש לדבר בזה הרבה עיין באליה רבא ומיהו אין למחות ביד הנוהגים להקל דהוא בכלל עת לעשות לה׳ שקשה לכל ציבור לכתוב נביאים בקלף כדין אבל בוודאי הציבור שהיכולת בידם יש להם לכתוב נביאים כדין ובפרט בימינו שמפזרים הרבה כסף על תכשיטי בהכ״נ שאין נחוץ וכונתם לשם שמים כדי לקיים "זה אלי ואנוהו" בוודאי מצוה להתנאות בכתיבת נביאים הקדושים.
...however one should not protest those who are lenient for it is within עת לעשות for it is hard for every community to write Neviim on Klaf as proper, but certainly a community which has the ability should write Neviim as proper, and especially in our days when lots of money is strewn on ornaments for the synagogue which aren't of pressing need and their intention is towards heaven to fulfill the verse "this is my God and I will glorify him" certainly it's a Mitzva to glorify with writing the holy Neviim.
