There are those who do not say "tzaischem" because of the idea that you do not want to sent the malachim away. One does not say to a guest as soon as they arrive "When are you leaving". SImilarly, when royalty arrives, it is an insult to ask when they plan to leave. One wants the "Queen" to stay as long as she likes.
Those who say tzaischem learn in a way similar to Yaakov and "Machanaim" of malachim. That is, as he left Eretz Yisroel the malachim of Chutz Laaretz greeted him and the Malachim of Eretz Yisrael left him. When he returned, the Malachim of Eretz Yisrael greeted him and the Malachim of Chutz Laaretz left.
Similarly, the Boachem are the malachim of Shabbos and the Tzeischem are the Malachim of the rest of the week. A person is never left unescorted.
Note that this is why we say Boachem before Tzaischem. If the Malachim of the rest of the week left first, there would be a fraction of time in which one was left alone.