Hello There,
This is a complex topic, as steppe nomad society is a very fluid entity, but I'll do my best to answer at least part of this. My knowledge of the Mongols is on the origin of Mongol-Speakers, not necessarily the Mongols themselves, but I can answer about the Huns.
Basically, the Huns and the Mongols, along with other steppe nomad societies, are usually traced by language. The Huns (Xiongnu) came possibly from the Yenisei-Ob Region in the Minusinsk Basin. They likely spoke a Yeniseian language when they expanded out into the Mongolian Steppes and formed the Xiongnu Empire. They then experienced a language flip when they assimilated the Turkic-speaking Dingling, Xinli, and Gekun peoples, retreating across the Altai and eventually migrating into Europe.
The Chinese histories trace Mongolic speaking peoples from the Donghu, a confederation destroyed by the Xiongnu around the time of the Qin Dynasty. The Donghu remnants in Manchuria spawned the Xianbei, the arch-nemesis of the Xiongnu, who like the Xiongnu would also experience a language flip to Turkic when they migrated towards Europe later on.
The Mongols were Mongolic-speakers. Mongolic and Turkic are closely related languages, along with Tungusic, but there is a lot of debate about how these languages evolved. It used to be Macro-Altaic theory, which suggested Uralic, Turkic, Mongolic, Tungusic, and Koreo-Japonic all evolved from a common language of Altaic, spoken in the Altai mountains before it began branching off possibly around 2000 BC. That's no longer thought to be accurate. Koreo-Japonic is no longer believed to be related while Mongolic-Tungusic and Uralic-Turkic are believed to have had a much more complex development.
The Eurasian Steppes are a relatively homogenous melting pot, which was dominated by Iranic-speaking nomads until the Huns came, and dominated by Turkic-speakers until today (barring Mongolia, mostly). By the time the Mongol identity and "ethnicity" (if you can really call it that) emerged, the Huns had been gone for 500-600 years. The Mongols originated South of Lake Baikal near the tributaries of the Amur River, East of the Orkhon, in the 12th Century AD, a completely different location than the Huns. They had genetic and anthropological similarities, but they were not the same people.
Sources:
Vaughn, Carter. "The Turks in World History."
Jankowski, Henryk. "Altaic Languages and Historical Contact." In "Current Trends in Altaic Linguistics: A Festschrift for Professor Emeritus Seong Baeg-in on his 80th Birthday", 523-545. Seoul: Altaic Society of Korea, 2013.
Di Cosmo, Nicola. "Ancient China and its Enemies: The Rise of Nomadic Power in East Asian History." 161-205.
Vovin, Alexander. "Did the Xiongnu Speak a Yeniseian Language?"
Half of it, the other half is buried on his Academia page: https://www.academia.edu/1804191/Xiong-nu_Part2