Absolutely no, many of us tend to confuse and also project that, then why do people tend to confuse?.
They tend to confuse because Buddha is mentioned as one of 23 leelaavtharas of lord vishnu in Bhagavatam. So who is Buddha mentioned in Vyasa Bhagahavatham then?. There is a prodigious difference between to Gowthama buddha who left his throne, wife, children and people to the buddha in bhaghavatham.According to Bhaghavatham vishnu takes incranation as Buddha as son of Anjani and jaani to help shiva in killing Tripurasus who become very powerful after doing veda adhyayana for so many years, and also getting powerful boons. Since tripurasuras start misusing their boons by teasing and killing normal people, vishnu incranates as Buddha who starts teaching Asuras a false propaganda about Vedas to misguide them. Asuras along with their wives(very powerful ladies, who always are reason for strength of asuras) fell to the hypnotism of buddha started believing in his teachings like ‘there is only one life to survive, and there occurs no karma siddantha, so we can start enjoying this one and only one life us, and rather spending time on reading Vedas we can attain moksha by dhyana’, since Asuras started believing in all these propaganda by him, in a very little period their strength and dharma decreases thereby it helps lord shiva to attack them and kill them with help of brahma who becomes chariot rider of him, and vishnu who becomes arrow that can kill Asuras. Even in ‘Apannivarika stotra’ Buddha is mentioned as ‘Daityastreemanabhanjine’ which says that he is the one who attracted Asura ladies towards his false propaganda.
There are many references to this incarnation in puranas of Bhaghavatham, vishnu, skanda and Vayu.
Here are few references of them,
Bhaghavatham, 10-40-22
namo buddhāya śuddhāyadaitya-dānava-mohine
mleccha-prāya-kṣatra-hantrenamas te kalki-rūpiṇe
Bhaghavatham, 1-3-24
tataḥ kalau sampravṛttesammohāya sura-dviṣāmbuddho nāmnāñjana-sutaḥkīkaṭeṣu bhaviṣyati
Buddah will take birth as son of anjana at start of kaliyuga to preach Ahimsa
Many indologists believe that Buddah mentioned in bhagavatam lived around 3400 BCE , which is around 5500 years from now.
Narasimha purana 36/29: ‘Kalau prapte yada buddho bhavannarayano prabhhu’
Nirnaya sindhu:’Jeysta shukla dwiteeyam Buddha jayanthi, poushya shuklasya saptamyam kuryate buddhasya poojanam’
Siddartha Gowthama buddha was from 500 BCE, he was born as son of mayadevi. He is also called as Sakyamuni. According to books of ‘Teravadi’ there are 6 kinds of Buddha’s Vipasya, Shikhi, krakkuchanda, konagaamara, and Maitreya. There is no relation between the Goutham buddha and the one mentioned in Amarakosha, but due to the non-voilence preachings of both people tend to confuse each other as same. Due to unstoppable and uncontrolled spread of non-voilence by Buddhism lead to raise of karma yogis which lead to destructions in the society. This situation was effectively controlled by Incarnation of shiva Adi shankara and thus lead to re establishment of veda samskruthi in Akanda Bharatha.
So Gowthama buddha is different from the one Buddha in Bhagavatam. And to add more to it leelavathara Buddha is never considered as one of dasavatharams in Bhaghavatham. From jaydeva dasavathara stotram to normal discussions on dasavatharams in hindu culture, balarama is considered as one of dasavatharams but not the Buddha.
My country, My dharma 