I'm growing both and even though my DH tree is not mature, I ate plenty of them last season and I was surprised. I personally enjoy the Dwarf Hawaiian eating experience but the unreserved recommendation given by many to grow it surprised me. The flavor is strong, sweet and delicious but somewhat gamey and definitely more fibrous than many eaters may be willing to accept. Dwarf Hawaiian has the stringiest, most fibrous texture of the modern mango cultivars I have tasted. That doesn't mean the Dwarf Hawaiian fiber is overwhelming, but it means that folks accustomed to smooth and fiber free mangoes are going to wonder how this mango was allowed to join the country club. The fruit is on the smaller side and Dwarf Hawaiian originates genetically from mangoes that do not enjoy a universal level of acceptance (Julie, Turpentine, etc). I think it fruits slightly earlier in the season than Pickering.
The Pickering fruit is larger and less fibrous. Pickering may have some coconut flavor but nothing too strong. Eating quality can vary, I have had outstanding Pickering fruit and quite a few that were only average in flavor and sweetness. Overall I consider Pickering to be pleasant, sweet, and delicious classic mango flavor that can rise even higher on a good day. The Pickering flavor will be more familiar and have wider appeal.
If Fruit Punch mango is a well-behaved and domesticated fruit, then Dwarf Hawaiian is the untamed primal experience straight from the jungle. Dwarf Hawaiian strikes me as closer to a wild type mango with small fruit, moderate fiber, and deep, earthy flavor. Dwarf Hawaiian shares some characteristics with Julie, Bombay and even the Turpentine mango. This is not a beginner mango with wide appeal, but something you learn to love further along the mango journey. I actually enjoy the body and bite of the flesh, it tastes strongly of very ripe nectarine with some added funk and a coconut background. If the Fruit Punch mango is Budweiser Beer, then Dwarf Hawaiian is a shot of Rye Whiskey.
In my yard, Pickering grows more vigorously and has required more pruning. If I had to plant just one, I think I would go with Dwarf Hawaiian because of the stronger coconut flavor and the negatives don't bother me. If I am recommending for someone else's yard, then I suggest Pickering.