@Elunelily The Steam version of Sims 3 does not require that the EA and Steam accounts be linked, and it doesn't work like Sims 4 at all. Even with your accounts linked, the EA App won't recognize the Sims 3 packs you own in Steam unless you register the content with EA.
In your position, I would uninstall the EA App version of Sims 3 and use Steam to install again. Most of us who have a choice prefer the Steam version because it doesn't require the EA App, or Steam for that matter, to be running; you can launch Sims 3 directly from its .exe after the first time. The Steam install also doesn't come with the pack selector that can forget your selections and load the wrong packs, or none at all, at inconvenient times. But it's your choice which app to use.
Either way, you'll need to register the product codes that Steam provides, one for the base game plus one for each pack. I believe they're under (Sims 3) Settings > Manage > CD Keys, or something similar, but let me know if you can't find them and I'll check my own Steam library. When viewing your EA App library, there's a button to add a game in the upper-right corner; that's what you use to register the codes.
Alternately, you could go to the Sims 3 Store site, sign in with your EA App account details, and use the "Register a Game" option at the bottom of the page to add each code.
The only way to buy SimPoints is through the EA App. But as long as your Steam product codes are registered to your EA account, any content you buy on the Store site will work in a Steam install as well as an EA App install. I personally have a Steam install and use plenty of Store content, as do many other players. And to be clear, since you're buying the points through the EA App, those points will automatically be credited to your account.
Once you have points on your account, you can buy whatever you want on the Store site. If you'd like, I can post some links that show how to get the most out of your points: there are sales and Daily Deals and Complete Your Set discounts that make the content much cheaper than its sticker price.
By the way, you should receive 1,000 SimPoints for registering each of the base game and World Adventures. If they're not automatically added to your balance on the Store site, click My Page and look for a Redeem Points button on the left side.
There are a few worlds that need to be purchased directly from the EA App. But these are also installed from the Store site and therefore also work in a Steam install.
