tefilla - Additions or exclusions of prayers in Siddurim

Someone just mentioned to me that there were differences of prayers in the Ashkenazic Siddur that are not in the Sefardic Siddur, without being specific. If this is true, could someone tell me which ones are in one and not the other? I always thought the only differences between siddurim were in the printed pronunciation of the transliteration and in wording changes, not the exclusions or additions of entire sections of the prayer.

Answer

Note: Not all Sefardim have the same Nussach, and not all Ashkenazim have the same Nussach. And this answer ignores the Chasidish [a.k.a. Sefard] Nussach which is a mix of the 2, at some level. And then there's the Nusacch from Teiman.

To answer your broad Ashkenazi vs. Sefardi Nussach:

The general structure of the Tefilot are the same - Shacharit, Mincha, [Mussaf] and Aravit.

Even in each of the above, the general structure is the same:

Shacharit: Birchot haShachar, Psukei D'Zimra, Shma and its Brachot, Amida, [Tachanun], [Kriat HaTorah], and the end pieces.

Mincha: Some preliminaries, Amida, [Tachanun], ending.

Aravit: Shma and its brachot, Amida, ending.

But once you get down to the details, the differences become apparent.

Here are a few:

Shacharit:

Sefardim will start with Tikun Hatzot and Ptach Eliyahu. The former is said at home by Ashkenazim (if at all) and the latter does not exists in Askenazi literature at all.

Sefardim say Hodu before Baruch She'amar; Ashkenazim first say Baruch She'amar.

There are minor differences in most Tefillot that are not direct quotes from scriptures - both in the Brachot around Shma and well as in the Amida.

The Tachanun is completely different, including the fact that Ashkenazim do Nefilat Apayim (leaning on their arms) and the Sefardim do not. The Sefardim preface Tachanun with the 13 Middot; Ashkenazim [mostly] do not. The Mizmor they say is different.

Mincha:

At Mincha the Sefardim start with Ptach Eliyahu which does not exist in the Ashkenazi literature.

Sefardim then say a long Seder Korbanot; Ashkenazim skip this or say VaYedaber, depending on custom.

The Tachanun is completely different, including the fact that Ashkenazim do Nefilat Apayim (lean on their arms) and the Sefardim do not. The Sefardim preface Tachanun with the 13 Middot; Ashkenazim do not. The Mizmor they say is different.

Sefardim add in a Mizmor (LaMenatze'ach) before Alienu; Ashkenazim do not.

Aravit

Sefardim preface Aravit with Shir HaMa'alot and Kadish. Most Ashkenazim do not.

During Aravit [usually called Ma'ariv by the Ashkenazim] the differences are mainly in the wording.

After the Amida the Sefardim once again add a Mizmor before Alienu. Some Ashkenazim will add 3 Mizmorim after Alienu.

On Friday night, the Kabalat Shabbat and Lecha Dodi is shorter by the Sefardim.

General

Kaddish has the same outline, but the wording is much longer by the Sefardim.

Selichot by the Sefardim is said the entire month of Elul and is almost unrelated to the [longer] version the Ashkenazim say during the week before Rosh Hashana, and on various days during the year.

Piyutim are added in abundance during special days by Ashkenazim, especially on Rosh Hashana and Yom Kippour. Sefardim do not have [most of] these additions.

As mentioned in the comments, the differences are numerous - you'll find them on almost every page of the Siddur.

That said, there are slight differences between different Sidurim; by both Ashkenazim and Sefardim [and Chassidim (and probably Teimanim)].