It depends on what your definition of "better" means. Paul has trained for far longer than Ash, but Paul is a snob. That is why Paul abandoned Chimchar... Because he was to "weak" for him. But Ash is definitely stronger willed, and most likely to become more powerful than Paul. But, of course, NOT a snob.
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Ash beats Paul in the episode that is titled "Battling a Thaw in Relations."
yes because in the sinnoh league semi-finals it was a long and hard battle but ash beat Paul with his infernape. Now ash and Paul are friends. Episode 650
He is so cold to Dawn because he likes her. Though Dawn likes Ash and maybe Kenny. Paul probably hates Ash because he gets to travel around with Dawn. Though, she doesn't like Paul for being cruel to Ash and Pokemon.
because he is secretly in love with ash
Yes and Ash owns him! Ifernape vs Electivire. Infernape FTW!