What is the reason why some people do not recite Hallel on Yom Ha'atzmut but recite it on Yom Yerushalayim? I've been at minyanim who practice this. Being that reciting Hallel on these days follows a certain point of view regarding the state of Israel, why would some people recite it on one but not the other?
While this is far from a "mainstream" concept, as far as I know, a number of yeshivot and Orthodox shuls in my mainly Orthodox neighborhood follow exactly this custom. (Do you live in NYC area, BTW?)
The reasoning is that they don't support the current Israeli secular government, and feel that the founding of what is now Israel was based on mainly secular concepts. (This reason partially explains why neither the shul nor yeshiva displays an Israeli flag or recites "Hatikvah" at social events.) However, all of them acknowledge the significance, holiness and importance of Yerushalayim, particularly, the Kotel. They all agree that the most significant accomplishment of the 6 Day War was regaining access to the Kotel, and that is something to say Hallel for.
I surmise that part of their thinking is making some sort of "political" statement, but it's not obvious to me what it would be and for what purpose.
Originally the Chief Rabbinate of Israel ruled that one should say Hallel without a bracha on Yom Haatzmaut ("The Chief Rabbinate ruled to recite Hallel without a blessing." http://hirhurim.blogspot.co.il/2007/04/hallel-on-yom-ha-atzmaut.html?m=1; "Because of all, or some, of these concerns, the Chief Rabbinate’s Council originally prescribed that one recite the Hallel without a blessing during the morning prayers of Yom HaAtzmaut." ph.yhb.org.il/en/05-04-07) and with a bracha on Yom Yerushalayim ("The Chief Rabbinate, in a ruling signed by Chief Rabbis Isser Yehuda Unterman and Yitzchak Nissim, along with the renowned R. Shlomo Yosef Zevin and R. Shaul Yisraeli, ruled that Hallel on Yom Yerushalayim should be recited with a blessinng." http://etzion.org.il/en/celebrating-yom-ha-atzmaut-and-yom-yerushalayim) This is because Yom Yerushalayim is more clearly a miracle as a definite military victory occurred then, whereas Yom Haatzmaut was merely the day of a declaration. Perhaps some people extended this reasoning.