The Santa Ana winds, locally called a "sharav" or "hamsin", are a spring phenomena and usually only last a day or two. Otherwise the weather will be good - hot but not yet the very hot summer temperatures. Nights will still be cool, especially in Jerusalem On the other hand, you will run into the holidays in September: New Years on Sept 5-6 and Yom Kippur on the 14th. No public transport on these days, lots of tourists and locals on vacation, crowded sites and high prices. I vote for June !

I went in late May/June for 2 weeks in 2007. I am very sensitive to heat, but I did fine in that timeframe. It was warm in the afternoon and I specifically remember being too hot at the top of Masada, I really wanted to sit in the shaded area on top and not walk around too much, but otherwise I did fine in the heat there.

I did not go to Eilat or Petra so I can't speak to the weather there.

I would choose June. We were there the middle of May and weather was perfect. In fact, in Petra we were a little chilly in the AM.
The holidays in Sept. could interfere with your travels.

We are planning to go in 2014, the High Holiday are later in September. So our options are June 5 - 23rd or 24th OR Aug 29-Sept 14 (I can make these dates a day or two earlier or later).

So would you still pick June?

I would go in June. August is very hot, and the end of August/ beginning of September will still be hot, especially in Eilat and Petra, and Tel Aviv can be humid in the summer.

I was once in Eilat in August where it was 113 degrees in the shade. Its a dry heat but you still bake.

If you can go to Eilat and Petra in the early part of June the weather should be not as hot as later towards the summer.

Thanks much.