The mishna (Menachos 68b) compares the minchas haomer which is offered on Pesach to the shtei halechem which is brought on shavuos.
"Haomer ha-ya matir b'medinah ushtei halechem b'mikdash - the omer would permit the new grains to be used outside the mikdash, whereas the shtei halechem allowed these grains to be offered in the mikdash." Both offerings, as we are taught in parshas Emor (Vayikra 23), permit the use of the new grain.
Although both menachos are mentioned in parshas Emor, the Torah in parshas Pinchas distinguishes between the omer and shtei halechem. Parshas Pinchas, which enumerates the korbanos haregel, the festival offerings, includes the shtei halechem but omits the omer. Indeed, in tefillas Mussaf on Shavuos we mention the shtei halechem. On Pesach, however, we don't include the minchas omer in our shemoneh esreh.
These distinctions point to a fundamental difference between the shtei halechem and the omer. Whereas the shtei halechem are included in the korbanos haregel of Shavuos, the omer, although offered on the second day of Pesach, is not an inherent part of the korbanos haregel of Pesach.
This essential difference is indicated in the Rambam as well. The Rambam (Hil. Klei Hamikdash, chap. 4) describes the division and rotation of the Kohanim who serve in the beis hamikdash. Each mishmar would serve for a week and was entitled to various parts of the korbanos. The korbanos haregel, however, were divided among all the Kohanim. When the Rambam delineates the korbanos haregel, he includes the shtei halechem but omits the omer. Although the shtei halechem was shared by all the Kohanim, the omer was given to the mishmar hakavu'a - the mishmar of that week.
The difference is also evident from the gemara in Eruchin which discusses the obligation of reciting Hallel on yom tov. Why do we recite a complete Hallel on chol hamoed Sukkos, but only a chatzi Hallel on chol hamoed Pesach? The gemara assumes that on each day of Sukkos different korbanos were offered, unlike Pesach, she'ein chalukin b'korbanoseihem. On the second day of Pesach in Eretz Yisroel only chatzi Hallel is recited despite the fact that the omer was offered on this day! Apparently, since the omer is not found in parshas Pinchas and is not counted among the korbanos haregel, only a chatzi Hallel is said.
Rav Betzalel Zholte explains that this distinction can be used to resolve a further difficulty in the Rambam. The gemara (Menachos 83b) explains that although ideally the shtei halechem and the omer are brought from the new grain; if no new grain can be found, grain which grew during the previous year can be used. Seemingly this statement is applicable to both the omer and the shtei halechem. The Rambam, however, cites this halacha only regarding the shtei halechem (Hil. Temidim U'mussafim, chap. 8) and not earlier (chap. 7) when discussing the omer. Apparently yashan, last year's crop, can only be used for the shtei halechem and not for the omer! Rav Zhotle suggests that the need for chadash, new grain, is rooted in the fact that the omer and shtei halechem serve to permit the use of the new grain, "Haomer ha-ya matir b'medinah ushtei halechem b'mikdash", only chadash, the new grain, can function as a matir. Accordingly the omer, which served as a matir, can only be brought from chadash. The shtei halechem, however, function as a matir as well as korbanos haregel. This second element, korbanos haregel, allows the shtei halechem to be brought from yashan if no chadash can be found.