Jordan and Israel are not the best friends in the world but both countries signed a peace agreement some decades ago so, surprisingly, travel to Jordan from Israel (and vice-versa) is pretty easy and straightforward.
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Currently, there are not any Jordan visa bans in place for any foreign citizen. A few exceptions include citizens of Lebanon and EU countries (except for Croatia), who may present a valid national ID card if arriving at Aqaba Airport on a direct flight from Brussels.
Independent Travel from Tel Aviv to PetraYou can do this by taking a bus or flight to Eilat and from there a taxi to the border crossing. Remember to organize your Jordanian visa before traveling. If you are traveling without a tour group you cannot get a visa at the border.
Jordanians do not differentiate between a Jew or a Christian or Muslim. All are same. Also as Patricia said, many Jews come by their cars to visit Jordan and no problems at all! On the contrary, the guest is always appreciated and respected more than local.
Venturing to Petra from Israel is possible by crossing one of the three borders. The border crossings in Eilat (Wadi Arava Crossing) and Beit Shean (Sheikh Hussein Crossing) issue visas on arrival. Some restricted nationalities are required to pre-arrange a visa, so be sure to check your requirements.
Petra is located in the southern part of Jordan, nearly 236 km away from Amman, Jordan's capital and business hub, where more than 40% of the population live.