Yes, there are differences, just like with mouthwashes.
What is important is the cleaning itself. The ingredients of a toothpaste help very little with oral hygiene. Some of them which are rich in fluoride help to prevent caries. There are others which are specifically for thermal sensation or to induce salivation.
When choosing a toothpaste, it is a good idea to receive a specialist’s advice. There are many toothpastes which can help a patient at a specific time, given the state of their oral health.

Related questions
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