It's not really balance that is causing this. Both games tend to be similarly balanced. They have annoyingly OP characters and horribly weak ones, with most picks sitting in a balanced state. The real reason why is the different design philosophies about counterpicking.
League is designed around weak counterpicks. While some champions may be strong or weak against certain enemies, you should ultimately have a fair chance of success in most every matchup. So long as you draft a balanced team comp, you should be able to respond to the enemy's team comp. However, this also means that champions are able to easily replace each other in the meta. If I need a siege mage, I can pick whoever is strongest between Lux and Ziggs and Xerath. If I need a tank, I can pick whichever one is strongest. Because the champions ultimately preform similar roles, I don't have much of a motivation to pick champions that aren't considered top-tier. After all, if Nidalee is a strong jungler and can do everything an assassin jungler could ever want, why pick someone who isn't Nidalee?
Dota instead has a lot of heroes who are very niche, but who absolutely excel in that niche. A good example might be Ancient Apparition. His ult is a global skillshot (with a good AoE) on a 40 second CD that does decent damage, completely prevents healing for 8-10 seconds (17 with aghs), and executes low health enemies for the duration of the debuff. This makes him a stupidly hard counter to healing heroes and heroes that get themselves low on health. Even if he wasn't particularly strong in general, he would probably be worth picking just for that role. That mechanic wouldn't be allowed in League.
There are also more heroes who have strong mechanics that are vulnerable to being counterpicked. A great example is Templar Assassin. One of her abilities gives her a shield that completely negates the next 7 instances of damage. This can let her take huge nukes to the face without taking any damage, but it also means that if she gets hit with a rapid DoT or something else that does many small instances of damage instead of one large blast, then her shield can be completely worthless. She gets a strong mechanic, but that mechanic has a weakness that you can only exploit with picks. If you don't counterpick TA, you're at a major disadvantage, and you'll need to have your teammates counter her or pick something that supplements your pick.