So most straightforwardly, the difference is that they're two unrelated dynasties—the first Safavid Shah, Isma'il, was not a descendant of Timur, the founder of the Timurid dynasty. Timur's sons and grandsons ruled in Eastern Iran for about a century, and then after the Safavids took over, a Timurid invaded India and founded the Mughal dynasty. There's effectively NO difference between the Timurids and the Mughals, other than the location they were operating in. (And obviously that location would eventually cause major changes in their ethnicity, culture, etc.) It's all just one family, tracing its legitimacy from the same source, which is descent from Timur the world-conqueror, with both the prestige he got from conquest, and the borrowed prestige he got from marrying into the family of Genghis Khan. Here's a genealogical chart made under the Mughals to show off their descent from Timur.

The Safavids, on the other hand, were on Shah Isma'il's paternal side the hereditary leaders of a Sunni order of Sufis, which is one of the more bizarre origin stories I'm aware of for an imperial Shi'ite dynasty. Also in the genealogical mix were the ruler of the Aq Qoyunlu Turkmen, Uzun Hasan, and the princess of Trebizond, who were the parents of Shah Isma'il's mother, incongruously named Martha. Shah Isma'il went on a campaign of military conquest that would ultimately result in his descendants ruling over a country with borders very similar to the borders of modern Iran. The Timurids, on the other hand, had mostly clustered in the east by the second half of the 15th century, with unrelated Turkmen dynasts ruling in the west.

Beyond just being two different families, there are other major differences between the Timurids and the Safavids, although their courts had a lot of continuity between them in terms of visual arts, literature, and culture. The most important difference, especially considering its impact on subsequent history, is that the Safavids were fervent Shi'ite partisans who declared the official religion of Iran to be Twelver Shi'ism. Before the Safavids came to power, Iran was a majority Sunni country, but generally practiced a fair amount of religious tolerance and pluralism; there wasn't a state religion in any real sense, and the Timurids, who were mostly Sunnis themselves, didn't tend to concern themselves too much with what their subjects believed (as long as they didn't cause trouble in other ways) (which they frequently did). Shah Isma'il and his descendants changed that unilaterally, and the impact on history has been profound. The entire conflict you see today between Iran and Saudi Arabia notionally derives from this fracturing of the Sunni consensus. (I would argue that this is actually historically false, and that the underlying issues are and always have been political rather than religious, but both sides disagree with my point of view on that.) According to most people in the Middle East today, the sectarian change instituted by the Safavids is the source of Iran's modern conflict with the Sunni world.