A major difference between 5s and 3s is how much lane control you need to win the game. To summarize: for 3s you want high dps champions with a lesser focus on team value, and in 5s you want high lane control champions with great team value.
In 5s you value champions that can hold their own lane well, because teammates will leave their lane open for minutes while they travel to help you. In 3s you value champions that can help other lanes and contribute lots to ganking, because traveling lane to lane takes very little time.
Champions like Heimerdinger and Soraka do well in 5s because they both have great lane control. Heimerdinger's turrets and passive heal allows him to push a lane very well and stay in good shape, meaning he doesn't need to leave the lane for a heal and spend a minute running back out. Soraka can heal herself and replenish mana, so she almost never needs to leave a lane.
However, in 3s they are less handy because they are squishy and susceptible to being ganked. They also provide little value when it comes to ganking in other lanes because they are not damage-per-second champions. To contrast Heimerdinger and Soraka, Tryndamere is a champion who shines in 3s. He can quickly travel from one lane to another by spinning over walls, and his dps makes ganks easy.
Another thing to note is that you require team composition in 5s, as any one champion can be focused down by 5 others at a time without a team backing them up. In 3s it's less of a problem because champions last longer in team fights as less damage is going around. Tryndamere in 5s can be crowd controlled by 5 other champions and he provides no aid to his teammates. However, in 3s he can solo two champions and likely win.