To summarize about VIDEO stutter (I assume that's the issue).
1) GPU decode - or "hardware acceleration" helps a lot. Your old laptop appears to only support MPEG2 (i.e. DVD). The media player must also be set to use this or it will simply use the CPU.
2) CPU decode - with no GPU decoding the CPU does all the work. If not fast enough your video will stutter.
3) Media Player - they vary in efficiency. It's generally the DECODER they are using and the settings.. K-Lite for example works very well. For some laptops this may make the difference between stutter and no stutter
4) Web browser - same as media player, though opening pages will eat up CPU cycles and increase the likelihood of stutter. No hardware acceleration from your GPU that I'm aware of on the T5800 since that's usually H264.
5) NETWORK - aside from CPU/GPU, you can also stutter or at least have a DELAY due to network bandwidth. That happens to me all the time with Youtube however my NETFLIX account rarely has issues.
NEW COMPUTER->
There ARE some weaker computers that can be insufficient even though they are brand new. Some of the weaker AMD or Intel CPU's may at times have issues especially with other tasks running.
Avoid Pentium/Celeron and weaker AMD. In fact, I'd suggest any i3-4xxx or better.
(Some AMD CPU's run at 1.4GHz and only have two cores. There's a HUGE difference in processing potential between that and a modern i3 which has hyperthreading, and has a Turbo of 2.4GHz.)
THIS APU has four cores, and the Intel has TWO physical cores with hyperthreading:
A4-5000->
http
/www.cpubenchmark.net/cpu.php?cpu=AMD+A4-5000+APU
i3-5010u->
http/www.cpubenchmark.net/cpu.php?cpu=Intel+Core+i3-5010U+%40+2.10GHz
Each core for the APU has a score of 586, whereas the Intel has a score of 1177. So each physical core of the Intel is 2X as powerful approximately which is especially important if a program has code that can only run on one thread (can't be split between CPU cores).
The HT for the Intel is virtually like adding weaker cores, but what really happens is the same CPU can process a separate thread of data during the normally unused CPU cycles (due to delays in getting new data). It can add up to 40% performance boost but that depends on how well threaded the program is.
*Basically look for a laptop with more than 1000 score for single thread and you're probably fine. Then concentrate on other aspects (like screen resolution, and most importantly the QUALITY of the laptop estimated from customer feedback).
GPU acceleration is important but it's best to have a good CPU to avoid any such issues. A modern i3 can likely decode a BluRay movie without GPU decode (all on CPU).
OTHER:
Task Manager (CTRL-ALT-DEL, or right-click "Start" to access) - this can monitor your CPU performance. You should monitor each core (not a single graph). Open it up and play a video. You should see the CPU usage spike. If it stays below 100% on all cores it shouldn't stutter (if it did it's not the CPU's fault). If one core hits 100% it may or may not stutter.
Anyway, you can determine if its the CPU that is your problem. My dad's laptop eight years ago couldn't play HD video. I got K-Lite later and it could with GPU acceleration. It can even play most HD now without GPU acceleration with K-Lite as the decoders have slowly gotten more efficient.
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