You paint a clear picture of what's going on, and that's helpful.
I think maybe you should look at this from your boyfriend's perspective. He has two children one that doesn't have a great shot in life - his parents aren't together, and he's struggling with adolescence, and he only gets to see him every other weekend. (I'm confused about that part actually because you say it's every other weekend and the son won't come over any other time and then you later say any time you want to do something with your son he's there too. Actually as he would be if you two were a fully functional family)
It's interesting you bring up that he treats your son like a pain in the *** step kid, and that's exactly the dynamic that's going on here. This 4 year old is yours, and the 13 year old is his. From your boyfriend's perspective the 4 year old has a cush life - both parents in the home, and a doting and effective mother.
I'm guessing your boyfriend feels guilty for the rocky start his son had, and feels guilty giving his youngest child a much better start in life, in full view of the 13 year old. And yes, the 13 year old does like to "do" stuff. Kids that age don't want to lay around the house and help clean out the garage, they want enjoyable adventures, which is what your boyfriend wants to provide in the very limited time he has him. That's understandable.
You say you can't even mention his son's name without getting yelled at - but look at it this way. If you said "wow, ______ did a great drawing! He's really doing well in art" or "Do you think ____ would just love this t-shirt?" do you think you'd get yelled at? It's likely you're saying negative things and your boyfriend is struggling with trying to defend his son.
I hope you work this out somehow. The only person you can control is yourself, and there are things you can do to make this better, like intentionally say nice things to and about his son.
I think your boyfriend sounds abusive. I think you need therapy to peel back the layers.
Is your 4 year old biologically your boyfriend's? I understood that to be the case in your question. I don't care how much guilt he feels over his 13 year old, he shouldn't treat your 4 year old poorly because the child has a better start. That's not healthy thinking.
Possibly look up the scapegoat/Golden child dynamic and how much damage it does to both children. You're already seeing it in the helpless 13 year old.
The reality is you are not a maid, and if something makes you uncomfortable it should matter!!! If he doesn't want to force the 13 year old to pick up after himself and he wants to be the bank, fine. He picks up after the 13 year old (not you) and he spends on the child out of his extra income, not family income. IE you pay mortgage/rent, electric, gas, cable, cell phones, internet, savings, 401k, college fund to the 4 year old, should be split equally between you. What is left over for him is where his money should come from for 13 year olds college fund and "fun times". If he is currently giving you more than half the bills but then blowing money on his 13 year old so he can be the cool parent then he is 1 disrespecting you, 2. Disrespecting his 4 year old, 3 putting the family at financial risk and 4 not being a reasonable and responsible good example to his children.
If you're religious, the Bible says to love your wive as Jesus loves his people. Does your husband do this??
(Also, if you saying anything negative about the 13 year old is causing him to yell at you, that is another red flag. It just isn't healthy. Teenagers can be little Jack wagons, they're notoriously hard to deal with. If you can't even vwnt to your spouse then k have to wonder how supportive of you he is. I sense from your post that your resentment is building. So your relationship is literally in danger. Time to seek real help before your resentment can't be fixed)