I almost passed this one... (also note that this is just my opinion)
Character design and personality
Erika - I'm not gonna lie, Erika's design is somehow ordinary, but still interesting. I feel like her design was inspiration for design of Aroma Lady Trainer Class. I'm not sure about her personality in games, but I hated her in the anime. But her episodes were also one of the most memorable ones.
Morty - Similar to Erika, Morty's design is ordinary, yet great. I prefer his HGSS design to be honest.
None of them really stands out, but I think I like Morty more, because of his HGSS design.
Pokémon
Let's talk about Pokémon team when you face them:
Erika - Erika's teams are really bad. Her Pokémon are really underleveled and have bad movesets. In LGPE, there are 7 moves distributed between 3 Pokémon. That's pretty boring for 4th Gym Leader if you ask me. At least, she has nice variety of Pokémon in each game.
Morty - Ghosts are interesting Type and it's shame that there weren't more in early Gens. Morty is 1st Gym Leader that specialize in them and sadly he only uses Gastly-line, even with Misdreavus being introduced in Johto (and being only Pure Ghost-Type at the time). His Pokémon are even more underleveled than Erika's, but they use actual strategy, like Mean Look + Curse or Hypnosis + Dream Eater combos.
Erika's teams are really weak. I would still say Morty has better team, even when it's made out of one evolution line.
Gym design and grunts
Erika's Gym fulfills its purpose. It's great garden and I probably like it the most out of every Grass-Type Gyms. But then, there is Morty's Gym, which has interesting and fitting puzzle in it in both iterations.
I can¨t decide in this one, to be honest.
Actual battle
Well, this battle is really hard to analyze. Morty's team has a lot of crippling moves and fast Pokémon, but Erika has some bulky hitters with powerful moves like Power Whip and Leaf Storm. To be honest, this battle could go both ways, but Erika's Pokémon has higher average level and could probably survive through all the Confuse Rays and Will-O-Wisps, especially if she is able to set up Sunny Day, as most of her Pokémon has Chlorophyll as ability.
In the end, I like both Erika and Morty, but I feel like Morty is more memorable of the duo in the games, while Erika is more memorable in the anime. But I gave my vote to Morty.