Is there a strict geographical definition on those names from a historian's point of view? Are there differences between the geographic definition e.g. is upper Mesopotamia included/excluded? I have the impression (not really sure) that while "Anatolia" includes eastern Turkey, Asia Minor doesn't extend that far. Strabo (2.5.24; 12.1.3) speaks of the region from Taurus and westwards.
According to the Online Etymology Dictionary, Anatolia is simply "the ancient name of Asia Minor".
However, here is some helpful background on the different origins of the two terms. In theory this could be seen as implying a different definition for Asia Minor, but in practice the terms are synonymous.
Asia Minor is a geographic region in the south-western part of Asia comprising most of what is present-day Turkey... It was called, by the Greeks, “Anatolia” (literally, 'place of the rising sun’, for those lands to the east of Greece). The name 'Asia Minor’ (from the Greek `Mikra Asia' - Little Asia) was first coined by the Christian historian Orosius (c. 375-418 CE) in his work Seven Books of History Against the Pagans in 400 CE to differentiate the main of Asia from that region which had been evangelized by the Apostle Paul (which included sites known from Paul’s Epistles in the Bible such as Ephesus and Galicia).
As for a precise definition, here's one from Encyclopedia.com:
Anatolia, also known as Asia Minor, is a large, mountainous peninsula of approximately 755,000 square kilometers (291,500 square miles) that extends from the Caucasus and Zagros mountains in the east and is bordered by the Black Sea on the north, the Aegean Sea on the west, and the Mediterranean Sea on the south. It comprises more than 95 percent of Turkey's total land area.
Wikipedia also treats the two terms as synonymous, but gives a slightly different definition:
The article goes on to explain that indicating the precise boundary of Anatolia is politically fraught in modern times due to tensions between Turkey and Armenia.
Although the answer is ticked as correct and helpful, it's completely wrong regarding naming origins. The name Asia was initially given by ancient Greeks to modern western Anatolia - later was used on about the same boundaries by the Romans for the roman province of Asia, which the Greeks called Asia or Asiane. The term Asia is of unidentified etymology. As mentioned correctly, the whole penisula was later called Asia Minor (Mikra Asia in Greek - Mikrasia is the term colloquially used in modern Greek, Mikrasiates the term for the Anatolian Greeks), in contrast to the rest of the continent which was named Asia.
The same name, Asia Minor, was used by medieval Greeks/Romans (medieval Greeks called themselves Romaioi, Romans in Greek. Byzantines is a neologism coined centuries after). One of the administrative-military provinces/divisions of the Eastern Roman empire (neologism, Byzantine Empire) was Eastern Division - Thema Anatolikon in Greek language. Thema Anatolikon covered the eastern part of Asia Minor, hence the name. From this term, Anatolikon, which appeared in medieval times, came the Latin deritive Anatolia. The greek term Anatolikon was also adopted by the invading turkic tribes as Anadolu, since they first invaded the Thema Anatolikon (Eastern Province/Division) of Roman Empire. Later it was used for the whole penisula.
Modern state of Turkey seems to extend the term to its whole Asian territory, not only the penisula. Probably it's a political decision, in order to avoid using geographic terms like Western Armenia, Northern/Upper Mesopotamia, Assyria, Kurdistan, inhabited in the past by Armenians, Assyrians/Syriacs and Kurds today.