Ace: People often use those words interchangeably, but yes there's a big difference!
When a character is described as asexual in fiction it's usually used as a shorthand for "aromantic asexual". A character who is asexual, and thus lacks an interest in sex, almost always also lacks an interest in any romance at all. This, however, is a simplified version of asexuality that isn't accurate to reality.
In real life, asexuals come in a variety of "romantic orientations" ranging from "heteroromantic" (romantically attracted exclusively to the opposite gender) and "panromantic" (romantically attracted to all genders), with "aromantic" (someone with no interest in romance) actually being the least common type. Most asexuals can fall in love and want to enter romantic relationships, though whether they have difficulties being in a relationship due to their sexuality is a different issue altogether.
Due to the prevalence of aromantic asexuals in media, this trope focuses on the ones who aren't aromantic. This in itself can lead to tension as most asexuals don't date other asexuals. How to make the relationship work, despite one partner not being asexual, is a common source of drama. Some couples are fine being a Chastity Couple while others do have sex.
Compare to Trans Equals Gay for other LGBT-related misconceptions and Puppy Love for a romantic relationship between people who are usually too young to feel sexual attraction. Related to Chastity Couple, which is about positive depictions of (usually non-asexual) couples that have a sexless relationship.
- It's implied in Sweet Blue Flowers that Akira might be asexual. This leads to complications with her lesbian love interest, but the manga's ending shows them still together in the future.
- In RWBY: Scars, Ruby's sister casually note that she's asexual in a discussion. Ruby also has a girlfriend, Penny. For that matter, Penny is a robot (albeit with a human soul) and it's never made clear before she dies whether she experiences sexual attraction either.
- The fanfic Ace of Hearts focuses on Keima Katsuragi coming to realise he is heteroromantic, but asexual.
- How To Be A Normal Person by T.J. Klune is a Queer Romance novel about a man named Gustavo meeting an asexual stoner named Casey.
- Perfect Rhythm by Jae is about a female pop-star named Leo who goes back to her home town and falls for a nurse named Holly, who is asexual.
- In The Posterchildren, it's revealed at the end of the first book that one of the main characters, Mal Underwood, is gray-asexual heteroromantic.
- April from Stranger And Stranger once dated a boy but dumped him due to her discomfort with sex and hearing about sex.
- Alice in the book Let's Talk About Love is biromantic and asexual, and has a girlfriend at the start of the book. After they break up, she falls in love with a man named Takumi who works at her library
- In the Wayward Children novel, Every Heart a Doorway, The Protagonist Nancy Whitman is asexual and notes that she feels no sexual desire for anyone. She does note though that she isn't aromantic though and develops feelings for one of her classmates, a transgender boy named Kade Bronson.
- Corin Cadence from Arcane Ascension is asexual and seems to mostly lean toward homoromantic. Overall he appears to be romantically attracted to any human that he can have an engaging and intelligent conversation with.
- In Robin Hood, Sam Troughton once stated that he envisioned his character Much as being asexual, presumably heteroromantic based on his Love Interests.
- Emmerdale: Liv Flaherty came out as asexual, spoke openly about it, with both her family and friends, and they're doing a storyline where she's going to go into a romantic relationship where she explains to the boy she's dating that she's asexual.
- In Sirens (US), Voodoo is asexual, but dates co-worker Brian, ostensibly out of pity and fully expecting that he'll get tired of not having sex and dump her. However, Brian stays completely faithful to her, turning down offers of sex from other women and not even engaging when Voodoo drags him to a strip club. Having realized that Brian wasn't just looking to "fix" her, Voodoo dumps him in hopes that he'll find someone willing to fulfill his sexual needs... only to realize afterwards that she's developed romantic feelings for him.
- Alvin in Backstage Pass, confesses to Sian that he really likes her and thinks she's attractive, but doesn't feel any of the feelings associated with arousal. They decide to explore their relationship further from that point.
- Virginia in Magical Diary is willing to have a relationship with the protagonist, but admits she isn't interested in doing more than kissing and cuddling with anyone and worries that this will eventually cause problems for them as a couple.
- The Urban Fantasy webcomic Ignition Zero stars a homoromantic ace named Robbie. He falls for another asexual man named Orson. Robbie explains being homoromantic to his roommate in an early page.
- Ace from Kimchi Cuddles is asexual and biromantic.
- The main lead's love interest in Rock and Riot, Carla, is a panromantic asexual.
- Carla Rutten of Dumbing of Age is an asexual homoromantic trans woman.
- In A-gnosis' comics on Greek myth, Athena is asexual and biromantic. She has some early insecurity about her orientation, but Aphrodite — who, as the setting's Love Goddess, definitely knows — reassures her that love can exist without desire as easily as the opposite.
- In Sister Claire, Catharine is a romantic asexual in a relationship with another woman.
- Chanel from the webcomic Rain is an asexual lesbian, and is in a relationship with non-asexual lesbian Maria.
- Shades of A (NSFW) is the story of Anwar, an asexual man who "doesn't want to do more than cuddle", entering a romantic relationship with Chris, a gay man who's active in the kink scene.
- Bojack Horseman: Todd comes out to his high-school sweetheart, Emily, in a diner during the Season Three finale. She accepts his asexuality and they date for a while, but they later end up breaking up because her own needs aren't being met. He also briefly dates an asexual woman, but they have little else in common and break up.