A woman who wishes to recite Taĥanun should not make an interruption by talking between Shemoneh Esrei and Taĥanun, because these prayers are more favorably accepted when connected to Shemoneh Esrei (SA 131:1; MB 1).
The Sephardic custom is to say Vidui (confession) and recite the Thirteen Attributes of Mercy before Nefilat Apayim, so that the atonement that they bring will lead to a climax with Nefilat Apayim (Kaf Ha-ĥayim 131:5). According to Ashkenazic and Yemenite (Baladi) custom, we only begin with Vidui and the Thirteen Attributes on Mondays and Thursdays.
It is said that the Thirteen Attributes of Mercy have great power of atonement for sins. By accepting the highest level of faith represented by the Thirteen Attributes, we connect to God in such a profound and exalted manner that our sins become marginal and external to us in comparison to inner devotion to God and His actions. This is the source of the atonement, and this is why we constantly repeat the Thirteen Attributes in Seliĥot and on Yom Kippur.
The Thirteen Attributes are considered a sacred matter (davar she-bikdusha) and therefore requires a minyan (Peninei Halakha: Prayer 21:5). Therefore, a woman praying without a minyan who wishes to recite Taĥanun omits the Thirteen Attributes.
In Sephardic custom, Tehilim 25 is recited during Nefilat Apayim, whereas Tehilim 6 is recited according to Ashkenazic rite.
On Mondays and Thursdays, additional Taĥanun prayers are recited, since those days are days of Divine grace when prayer is more accepted. The extra passages are recited while standing (SA and Rema 134:1). The text of Ve-hu Raĥum was compiled by three elders who were exiled from Jerusalem, as stated in the works of Rishonim (Abudraham, Raavan, Manhig, Kol Bo §18). Therefore, the distinctions in its wording between the various customs are slight, except that Sephardim add more Taĥanun prayers before it and Ashkenazim add more Taĥanun prayers after it.
Another difference is that in the Sephardic and Ĥasidic rites, the additional Taĥanun prayers of Mondays and Thursdays are recited after Nefilat Apayim, whereas in the Ashkenazic rite they are recited before Nefilat Apayim.