I didn't know Buddhists worshipped Buddha as a god. He certainly claimed he was no deity at all. Buddhist theology defines a "god" as a person living in the higher spiritual realms, composed of a spiritual body, and having a limited (even if eon-long) lifetime. Buddhist cosmology posits that many, if not all, humans have, at some point, taken rebirth as a "god", due to their practice of dharma, or "what is right". The Buddha's previous lives included many lifetimes as a "god".
However, it is true that the Buddha's final lifetime, was spent as a human. A human can realize the end of suffering ("nirvana"), whereas a "god", in general, can't. It would be more accurate to say that Buddhists worship/venerate Shakyamuni as a Buddha; to worship/venerate him as a mere "god" would be to insult him.
What many people dont understand is that Buddhism is a very diverse philosophy with many schools of thought (denominations). The most authentic Buddhists dont worship a God. There are some Buddhists who are Polytheistic such as those who worship Hindu Gods or those who worship Shinto or Chinese Gods in addition to those who practice ancestor worship. There are also some Buddhists who have deified the Buddha and worship him, but thats not the mainline teaching of Buddhism.
It's a highly dubious enterprise to distinguish between a supposedly pure Buddhism and a corrupted Buddhism. It is certainly "mainline" among Mahayana Buddhists to see the Buddha as the eternal, underlying reality (and thus, comparable to concepts of God). It is also very "mainline" for Hindu, Chinese, and Shinto deities to be revered by Buddhists as "dharma protectors" or even as emanations of Buddhas. For example, the Hindu goddess Saraswati is revered in Tibet as a bodhisattva. Rather than completely supplanting local cults, the usual Buddhist practice was to bind the gods as protectors of the dharma, and this is a very orthodox Buddhist practice.
Also, while there are many different strands of Buddhism, I think it's safe to say that they are far more unified in fundamental beliefs than the various Hindu or Christian sects.
Tell that to the Dalai Lama or the Hardcore Theravada Monks in Thailand, Im sure they wont agree that worshiping these deities or the Buddha would be acceptable according to what Buddha preached. However, these practices are well accepted by a very large number of Buddhists.
If by "worship" you mean "honor, venerate", then the "worship" of the deities (powerful spiritual beings who can inspire one to practice dharma) is part of traditional Theravada practice.
The Theravada scriptures refer to the different subjects of meditation that are possible, the "ten recollections
". In addition to meditating on the Buddha, or the Dhamma, or the Samgha, one can meditate on the Deities (the "Devas"), the sixth recollection. The Devas achieved their status because of their faith/conviction, virtue/ethics, learning/knowledge, generosity/charity, and discernment/wisdom. Recollecting, remembering, meditating on, and venerating the Devas can inspire one's own practice.
The Buddha told Mahanama
: "Mahanama, you should develop this recollection of the devas while you are walking, while you are standing, while you are sitting, while you are lying down, while you are busy at work, while you are resting in your home crowded with children."