To be fair, both Tiger Udyr and Phoenix Udyr have their own merits.
Tiger Udyr has the advantage of having much more single-target damage and giving him access to AD items that he otherwise wouldn't have with Phoenix. He can use a large variety of items with Tiger max, including The Black Cleaver, Sterak's Gage, and Trinity Force, not counting any of the heavier AD items that are made for AD champions that build a lot more AD than he does. In the jungle, Udyr can afford to build Warrior enchant, which he cannot do with Phoenix Udyr, in addition to Cinderhulk and Devourer. In addition, Tiger Udyr also has faster turret taking potential because of the extra attack speed on Tiger Stance.
Phoenix Udyr has the advantage of making him more AoE-focused and is less reliant on items. The base damages on Phoenix Stance are high enough that he can afford to build almost full tank and still be capable of dealing decent damage, which is important because Udyr doesn't have too many good offensive items to build to power up Phoenix Stance. You can honestly build full tank as Phoenix Udyr and do fine, but you can't do that as Tiger Udyr.
One of the few and more interesting options Udyr has as Phoenix Udyr is Wit's End, which is one of the few items that will give him a relevant enough damage increase as Phoenix Udyr without depriving him too much of his defensive stats, and Phoenix Udyr can actually make a Wit's End work. In the jungle, Phoenix Udyr gets better synergy with Sated Devourer than Tiger Udyr, and has better synergy with Cinderhulk unless you build Titanic Hydra on Tiger Udyr (where the synergy is debatable), not to mention that Udyr's clears are slightly faster with Phoenix than with Tiger. Phoenix Udyr also has faster wave pushing potential than Tiger, which allows him to make a lot more moves around the map.
Personally, I like both styles and use both styles when I play Udyr either in top or jungle because of their different strengths, mostly because of their damage types. I use Tiger when my team has a lot of magic damage and Phoenix when my team has a lot of physical damage. I recommend trying out both styles and experimenting with both and see what you like with both, and learning both so you have that versatility between physical single-target damage and magic AoE damage.
Also, I recommend checking out the player Trick2G on Youtube. He's one of the best Udyr mains and is easily the most well known Udyr main around, and he plays a lot of both Tiger and Phoenix Udyr in both top and jungle, and you might learn something from his videos. Just don't learn his disrespect habits or try to copy his great escapes.