The Mishnah discusses in detail the laws of Pesach Sheini: who is obligated, what differences exist from the first Pesach (no prohibition of chametz in one's home, eaten for only one night rather than two), and how it serves as a genuine makeup offering rather than a lesser substitute.
מִי שֶׁהָיָה טָמֵא אוֹ בְדֶרֶךְ רְחוֹקָה וְלֹא עָשָׂה אֶת הָרִאשׁוֹן, יַעֲשֶׂה אֶת הַשֵּׁנִי. שָׁגַג אוֹ נֶאֱנַס וְלֹא עָשָׂה אֶת הָרִאשׁוֹן, יַעֲשֶׂה אֶת הַשֵּׁנִי. אִם כֵּן, לָמָּה נֶאֱמַר טָמֵא אוֹ שֶׁהָיָה בְדֶרֶךְ רְחוֹקָה, שֶׁאֵלּוּ פְּטוּרִין מֵהִכָּרֵת, וְאֵלּוּ חַיָּבִין בְּהִכָּרֵת: מַה בֵּין פֶּסַח רִאשׁוֹן לַשֵּׁנִי, הָרִאשׁוֹן אָסוּר בְּבַל יֵרָאֶה וּבַל יִמָּצֵא, וְהַשֵּׁנִי, מַצָּה וְחָמֵץ עִמּוֹ בַּבָּיִת. הָרִאשׁוֹן טָעוּן הַלֵּל בַּאֲכִילָתוֹ, וְהַשֵּׁנִי אֵינוֹ טָעוּן הַלֵּל בַּאֲכִילָתוֹ. זֶה וָזֶה טָעוּן הַלֵּל בַּעֲשִׂיָּתָן, וְנֶאֶכָלִין צָלִי עַל מַצּוֹת וּמְרוֹרִים, וְדוֹחִין אֶת הַשַּׁבָּת:
One who was ritually impure or on a distant journey and did not observe the first Pesaḥ by participating in the offering of the Paschal lamb on the fourteenth of Nisan should observe the second Pesaḥ by participating in the offering on the fourteenth of Iyyar. If one unwittingly forgot or was prevented due to circumstances beyond his control and did not observe the first Pesaḥ, he too should observe the second Pesaḥ. If so, that the second Pesaḥ is observed even by someone who forgot or was prevented from observing the first Pesaḥ, why is it stated in the Torah that the second Pesaḥ is observed only by one who was ritually impure or on a distant journey? These cases were specified in order to teach that these two groups of people are exempt from karet if they do not observe the second Pesaḥ, but those who were not ritually impure or on a distant journey are liable to receive karet, as the Gemara will explain. What is the difference between the Paschal lamb offered on the first Pesaḥ and the Paschal lamb offered on the second Pesaḥ? On the first Pesaḥ, at the time of slaughtering the Paschal lamb, it is prohibited to own leavened bread due to the prohibitions: It shall not be seen, and: It shall not be found. And on the second Pesaḥ it is permissible for one to have both leavened bread and matza with him in the house. Another difference is that the Paschal lamb offered on the first Pesaḥ requires the recitation of hallel as it is eaten and the second does not require the recitation of hallel as it is eaten. However, they are the same in that the Paschal lambs sacrificed on both the first and second Pesaḥ require the recitation of hallel as they are prepared, i.e., as they are slaughtered, and they are both eaten roasted with matza and bitter herbs, and they override Shabbat in that they may be slaughtered and their blood sprinkled even on Shabbat.