Will I notice a difference going from 16 GB of RAM to 32?

Speaking specifically for WoW, mainly SoD.

Would I notice a difference or should I save my money?

The specs say 16 GB is the Max but it’s actually 32.

Processor AMD Ryzen™ 7 4800H (8C / 16T, 2.9 / 4.2GHz, 4MB L2 / 8MB L3)
Graphics NVIDIA® GeForce GTX 1650 Ti 4GB GDDR6
Chipset AMD SoC Platform
Memory 1x8GB SO-DIMM DDR4-3200
Memory Slots Two DDR4 SO-DIMM slots, dual-channel capable
Max Memory Up to 16GB DDR4-3200 offering
Storage 512GB SSD M.2 2242 PCIe® 3.0x4 NVMe®
Storage Support Up to two drives, 1x 2.5" HDD + 1x M.2 SSD • 2.5" HDD up to 1TB • M.2 2242 SSD up to 512GB • M.2 2280 SSD up to 512GB
Card Reader None
Optical None
Audio Chip High Definition (HD) Audio
Speakers Stereo speakers, 1.5W x2, Dolby® Audio™
Camera 720p with Privacy Shutter
Microphone 2x, Array
Battery Integrated 45Wh
Power Adapter 135W Slim Tip (3-pin)
DESIGN
Display 15.6" FHD (1920x1080) IPS 250nits Anti-glare, 120Hz
Touchscreen None
Keyboard Blue Backlit, English (US)
Case Color Onyx Black
Surface Treatment IMR (In-Mold Decoration by Roller)
Case Material PC + ABS (Top), PC + ABS (Bottom)
Dimensions (WxDxH) 359 x 249.6 x 24.9 mm (14.13 x 9.83 x 0.98 inches)
Weight 2.2 kg (4.8 lbs)

For WoW? No. Hell, for most circumstances - especially gaming - you won’t notice even a little difference. You would be able to check the Task Manager and see that it’s using over 16GB at times, but that’s more the memory manager being lazy and not unloading things it could than anything actually meaningful.

Note that it can even slightly reduce performance if you’re going from two sticks to four. Most memory controllers will need to slow down the command rate from 1t to 2t when all of their banks are full to maintain stability - especially when using DIMMs (double-sided or double-stacked) instead of SIMMs.

It’s perhaps worth mentioning that there’s still very little appreciable difference in gaming performance between 8GB and 16GB for the most part. However it’s also getting increasingly more difficult to obtain 4GB SIMMs, and going down to single channel (ie 1x 8GB) will have a noticeable, negative impact on performance.

The only games I can think of that you would see a noticeable difference in performance with more RAM are games with massive maps that must remain loaded at all times.
For instance, when I play Galactic Civilization III with a ludicrous size map and as many bots as possible. Memory is consumed and each turn takes a few minutes to process. And that is with 32GB RAM. With some spare RAM, I tried it with 64GB and each turn took less than a minute.

For WoW all by itself, most likely not - you can even get by with 8GB, but that’s the minimum. I remember reading an article some years back that did comprehensive tests playing the game on different settings and talking about how the required specs can change to maintain decent performance.

It also depends on your computer usage behavior while you play WoW.
-Tacking on add-ons increases its RAM usage.
-Do you have browser tabs open in the background on third-party sites to look up WoW-related content or gear-sims while you play?
-Do you run Discord or WoW raid loggers

And of course, you don’t want to run out of physical RAM and start tapping into the swap file. Then your system would slow to a crawl.

I often look at Windows’ task manager to get an idea of my average RAM usage to make an informed decision.