The Mishna in Masechet Berachot (54A) establishes that when a person sees a bolt of lightning or hears a clap of thunder, he recites the Beracha, "Oseh Ma'aseh Bereishit," and this Halacha is codified in the Shulchan Aruch 227:1. However, the Taz (commentary to the Shulchan Aruch by Rabbi David Halevi, Poland, 1586-1667) observed the widespread practice to recite upon hearing thunder a different Beracha, that of "She'kocho U'gevurato Malei Olam" (Shulchan Aruch ibid.) The force of Hashem certainly manifests through the thunder.
This is, indeed, the accepted practice, to recite "Oseh Ma'aseh Bereishit" upon seeing lightning, and then "She'kocho U'gevurato Malei Olam" upon hearing thunder. This is also the ruling of Chacham Ovadia Yosef in Halichot Olam Helek 2, page 186.
One must recite these Berachot "Toch Kedei Dibbur" within a second or two of seeing lightning or hearing thunder. If a person saw lightning or heard thunder and did not recite the Beracha within a second or two, he may not recite the Beracha until he sees another bolt of lightning or hears another clap of thunder (ibid page 187).
One recites these Berachot only once during a storm; after a person recites the Beracha over lightning and the Beracha over thunder, he does not repeat these Berachot over subsequent bolts of lightning or claps of thunder. If the storm passes and then another begins, even later in that same day, then one recites the Berachot when he again sees lightning and hears thunder (ibid).
The Ben Ish Chai (Rabbi Yosef Chayim of Baghdad, 1833-1909) in Parashat Ekeb, Halacha 16, noted that the practice in Baghdad was to recite this Beracha without "Shem U'malchut," that is, without the phrase, "Hashem Elokeinu Melech Ha'olam." The Jews in Baghdad would recite upon seeing lightning simply, "Baruch Oseh Ma'aseh Bereishit," and upon hearing thunder, "Baruch She'kocho U'gevurato Malei Olam." The Sefer Petah Hadebir by Rav Hayim Pontrimuli in siman 227, s"k 3, explained this practice as based on the concern that people might not be aware of the requirement to recite the Beracha within "Toch Kedei Dibbur" of the lightning or thunder, and might mistakenly think that these Berachot are recited each time one sees lightning and hears thunder, even during the same storm. These misconceptions would result in the recitation of many Berachot Le'vatala ("wasted" Berachot), and the practice thus evolved to omit "Shem U'malchut" when reciting these Berachot.
The Ben Ish Chai further wrote that he posed this question to Rabbi Eliyahu Mani (Baghdad-Israel, 1824-1899), who responded that the communities in Eretz Yisrael likewise observed this custom, which is based upon the ruling of a work entitled "Petach Ha'devir." Indeed, the "Petach Ha'devir" noted the practice to recite these Berachot without "Shem U'malchut," and he suggested that it perhaps evolved due to the concerns mentioned above. To avoid the risk of multiple recitations of unnecessary Berachot, people relied on the position of the Ra'avad (Rabbi Avraham of Posquieres, France, 1120-1198), who held that all the Berachot mentioned by the Gemara in the ninth chapter of Masechet Berachot are to be recited without "Shem U'malchut."
Importantly, however, the "Petach Ha'devir" adds that people with Halachic proficiency, who fully understand the requirement to recite these Berachot within one or two seconds, and that they are not repeated in every instance of thunder of lightning during a storm, should recite the Berachot with "Shem U'malchut." This is, indeed, the ruling of Chacham Ovadia Yosef, that nowadays, when people are educated and have a general knowledge of basic Halachic guidelines, one should recite the full text of these Berachot, with "Shem U'malchut." (See Halichot Olam ibid)
Summary: When one sees lightning he must recite the Beracha, "Baruch Ata Hashem Elokeinu Melech Ha'olam Oseh Ma'aseh Bereishit"; upon hearing thunder, one recites, "Baruch Ata Hashem Elokeinu Melech Ha'olam She'kocho U'gevurato Malei Olam." One must recite these Berachot within a second or two of seeing the lightning or hearing the thunder, and each of these Berachot is recited only once during a storm.
