It is proper not to interrupt by talking between Shemoneh Esrei and Tachanun, because when the recital of the prayers of supplication is linked to Shemoneh Esrei, the prayers are more favorably accepted (Shulchan Aruch 131:1; Mishnah Berurah 1).
Concerning the nusach of Tachanun, there are differences between the various ethnic groups. The reason for this is that when most Jews lived in Israel or in Babylon, each person would recite prayers of supplication in his own tongue. Only later on, in the time of the Rishonim, when the exiles dispersed, certain set wordings took shape. Additionally, approximately four hundred years ago, some changes were made in Nusach Sephard, based on Kavanot HaAri.
According to Kavanot HaAri, it is customary to recite Vidui (confession) and the Thirteen Attributes of mercy before the Psalm of Nefillat Apayim, so that after achieving atonement from the recital of Vidui and the Thirteen Attributes, one arrives at the pinnacle of these prayers, Nefillat Apayim (Kaf HaChaim 131:5). According to the Ashkenazic and Yemenite (Baladi) minhag, we open with Vidui and the Thirteen Attributes only on Mondays and Thursdays, the days when we recite numerous prayers of supplication. However, on the remaining days, we say Nefillat Apayim immediately following the Amidah, because it is best to adjoin Nefillat Apayim to the Amidah as much as possible.
In Nefillat Apayim, according to the Sephardic Nusach, Psalm 25 is recited, whereas those who follow Nusach Ashkenaz and Sephard-Chassidi say Psalm 6.
On Mondays and Thursdays, additional prayers of supplication are recited, since those days are days of Divine grace when prayer is more graciously accepted. They are said while standing (Shulchan Aruch and Rama 134:1). The prayer “V’Hu Rachum” was compiled by three elders exiled from Jerusalem, as explained in the writings of the Rishonim (Abudraham, Ra’avan, Manhig, Kolbo 18). Distinctions in wording between the various customs are slight, except that the Sephardim add more prayers of supplication before it and the Ashkenazim add prayers of supplication after it.
Another difference is that in Nusach Sephard the additional prayers of supplication of Mondays and Thursdays are recited after Nefillat Apayim, whereas in Nusach Ashkenaz they are recited before Nefillat Apayim.
A person who practices according to one nusach and prays in a place in which most people are praying in another nusach is permitted to pray as he wishes. If he decides to follow his own custom, he should not make his different minhag noticeable. If the chazan is reciting the Thirteen Attributes, even one who does not have the custom to recite them must join the congregation. Someone reciting a longer nusach when the chazan starts saying Kaddish, must stop his prayers of supplication, respond to Kaddish, and continue on to the next stage of the prayer service. The reason for this is that the exact wording of the prayers of supplication does not prevent a person from fulfilling his obligation, and anyone who recited even a few prayers of supplication has already fulfilled his obligation. If he so desires, he may finish the prayers of supplication after the prayer service.