I pointed out one difference - very often with my cat, I'll be doing something sedentary like reading, and she'll come up and sit a foot or two away from me. She'll make no overt attempts to attract my attention, but I can reach over every so often and stroke her, and she'll enjoy this without asking for more.
A dog would never do that - sure, they may decide to lie down near you, but if you show them the slightest bit of attention they'll perk up and start trying to interact with you. It's flattering, but I personally find it a bit overwhelming, and I have a habit of avoiding initiating interaction with dogs unless I'm sure I can handle excited friendly dog overtures.
The thing is, when my cat sits near me and lets me pet her, this is affection. There's a stereotype that cats aren't affectionate, but they really are. They just have a quieter, more low-key style of affection than dogs do. (That is, when they don't just leap on top of you, which cats will often do as well, though dogs tend to jump at people more readily than cats.) You can see this clearly in the patterns of who the cat will and will not come near to. If a cat doesn't like you, they'll always be out of reach. Very often they'll be out of sight, too. Cats pay careful attention to a human's reach, and far out of reach, barely out of reach and in reach mean different things. (Sleeping barely out of reach seems to mean they like you but don't feel like getting petted, or if they're being active it means they want to lead you somewhere.) I see similar behavior in the nature show Big Cat Diaries, with leopards (who act a lot like huge housecats) choosing to sleep about a foot away from their half-grown cubs.
And it's not just cats and dogs that show this contrast. I've noticed a similar difference between autistic and non-autistic children. I often hang out in the same room as my parents, not necessarily interacting, but just being near them. In working with children, I've found often I can sit near an autistic kid and engage in occasional brief interaction while mostly just enjoying their company. I've even had autistic kids do the same back to me, shooting me brief glances or occasionally initiating a very brief interaction. (For example, with one kid I started playfully stepping on his feet in the pool. Every so often, when we weren't actively interacting, he'd step on my foot and give me a brief playful glance.) Non-autistic kids, whether they're neurotypical or have some other developmental disability, rarely let me get away with brief interactions - if I show them any attention, they want me to give them undivided attention, just like a dog does.
Which makes me think that people think autistics aren't affectionate for the same reason they think cats aren't - because both cats and autistics have a low-interaction style of affection that is marked by seeking out proximity with rare or no initiation of interaction, and a preference for brief interaction when you get it. I also wonder if the already low rate of initiation in autistic kids is, in some cases, even lower because they know that, like me with dogs and non-autistic kids, if they show the slightest bit of attention to the person they'll get more attention back than they really want.
If you have an autistic loved one and you think they don't care about you, pay attention to where they chose to be. Do they chose to be near you more often than could be explained by sheer chance? (They may not follow you from room to room, because autistics often have trouble shifting attention, but if they do follow you, even delayed, this is a clear sign.) Also try, when they initiate interaction, responding with a brief, low-intensity response and waiting for them to signal if they want more. And if they're not in the mood to interact, try just setting yourself up to be near them, while pretending not to notice them.
And remember that a different style of affection isn't necessarily better or worse. It's just that it can result in a communication barrier if it's not well understood. (There's a pop psychology theory floating around that there are 5 different love languages, and each person prefers to communicate and receive love in one particular language, causing problems if these don't match. While I'm not sure if there are specifically five love languages or if I like the five he's picked, I do know that the basic idea is right, and could be very useful to interaction between autistics and non-autistics.)