They're two different sets of psalms, sad psalms in Tikkun Rachel, hopeful ones in Tikkun Leah. On regular days both are recited. On days when mourning is inappropriate, only Tikkun Leah is said.
The names Rachel and Leah are used in their Kabbalistic sense, in which they refer to partzufim (faces or manifestations of God).(*)
In the Kabbalists' understanding of the rhythm of time, two related processes happen at midnight: the partzuf Leah enters into a state of union with the partzuf Yaakov, and the partzuf Rachel descends below the lower limit of the world of Atzilut. (**)
The ritual of Tikkun Hatzot is a way of acknowledging and participating in these supernal processes. The process involving the partzuf Rachel has characteristics of exile and separation, and so psalms of sadness are appropriate.
(*) The Ramhal provides a list of the 12 partzufim, including Rachel and Leah, at the beginning of Klalim Shniyyim:
http://www.hebrewbooks.org/pdfpager.aspx?req=51288&st=&pgnum=135. See the footnotes there for definitions.
(**) See, e.g, the introduction to Tikkun Hatzot in Siddur Hemdat Yisrael for a description of these two processes: http://www.hebrewbooks.org/pdfpager.aspx?req=53884&st=&pgnum=10
