Some people, with good reason, automatically assume that actions are allowed on Shabbos unless proven otherwise. Therefore, they see use of twist-ties as presumptively permissible. Others see the clear similarity between twisting a twist-tie and tying a knot, and assume the former is forbidden like the latter. Both are right but the subject is more complicated because a possible forbidden parallel to twist-ties can be found.
Twist-ties are not automatically forbidden. An argument can be made that twisting a twist-tie does not constitute tying a knot. While the ends of the twist-tie intertwine repeatedly, they are merely wrapped rather than knotted. Additionally, twist-ties are made to be wrapped and unwrapped. When twisting them, you are not creating a knot but using the object in its normal function.
On the other hand, the Rambam (Mishneh Torah, Hilkhos Shabbos 9:8) writes that someone who coils a rope violates a sub-category of tying a knot. Even though the rope has no knot, merely wrapping the strands in a way that they will remain together is forbidden. This would seem to also preclude uniting the two ends of the twist-tie into a coil, even if they subsequently split apart. Significantly, the Magen Avraham (317:20) and Mishnah Berurah (317:34) follow this Rambam.
One could counter, in turn, that coiling is only forbidden if it is permanent. If it is not, then there is no indication that it is even rabbinically prohibited. Or you could argue that the comparison with a rope fails because, with a twist-tie, the coil does not hold the ends together but rather the hardness of the material.
In practice, we find halakhic authorities on both side of the equation. Shemiras Shabbos Ke-Hilkhasah (ch. 15 n. 166) quotes R. Shlomo Zalman Auerbach as forbidding the use of twist-ties if you are sealing them for longer than one day. Orechos Shabbos (ch. 10 n. 51) quotes R. Yosef Shalom Elyashiv as ruling strictly, as well. R. Mordechai Eliyahu (Imrei Mordechai p. 182-183) and R. Ephraim Greenblatt (Rivevos Ephraim 3:552) also forbid their use (see also Ma’amar Mordechai vol. 1 no. 14).
However, Orechos Shabbos also quotes R. Shmuel Wosner and R. Nissim Karelitz as permitting the use of twist-ties. Similarly, R. Eliezer Melamed (Peninei Halakhah, Shabbos 13:16 n. 14) permits their use.
I learned an easy compromise between the two positions from my wife. Whenever she closes a bag with a twist-tie, she does not twist the two sides together but circles the twist-tie around the edge she wishes to close. In that way, she seals the edge without tying a knot according to any opinion. I am not sure if this practice was started to satisfy all halakhic authorities (she doesn’t consciously do it) but it is certainly a viable method to benefit from twist-ties according to all views.