I had a dear friend tell me, "Nerds are like firefighters. In an emergency, you all look alike..." :)
So, I've built dozens of PCs and rebuilt a handful of laptops. My builds have been historically AMD. Probably 80/20% AMD over Intel. The first AMD chip I bought was a 333MHz K6 chip, which is my way of saying I've done this for awhile.
For the casual user, there really isn't a notable difference. Windows will run Windows on either. Linux will run Linux on either.
It pains me to say it, but about 5 years ago (give or take) Intel took the lead in performance over AMD. AMD has yielded the high-end stuff to Intel and has placed bets on other areas.
Now I just bought an AMD A10 for my son, and it's great. AMD is still a good value. But if I had high powered, CPU intensive stuff to do I'd pay the extra $$ for Intel.
Longwinded way of saying - I don't think you care. Buy whatever floats your boat. If you're into Value, them AMD or an older Intel i3 or i5 chip. If you're into Horsepower, then Intel.
If you really want to know the differences, then we can start a long thread. :)
As for upgrading an old PC, yeah, I tend to agree. But more along the lines of it's just too expensive. Those components inside use technologies that have been surpassed - and upgrading a new PC is often more expensive than you'd expect.
Said another way, if you had wanted to bump the memory up back in 2009 or 2010, you could have paid $50. Now that very same increase might be $150 or $250 because those chips aren't in demand anymore.
If you're a casual user, if you're not a nerd (like me), then I'd say set your budget and shop for something within that budget.
Others will swear by brands (HP, Dell, Lenovo). I'll simply say I've used them all and have had lemons and gems in all.