Ignore the sequential speeds everyone focuses on.
The Achilles' heel of HDDs is their random small file access times. Because they need to reposition the read/write and wait for the proper sector of the platter to spin under the head for each small file, their small file random read/write access speeds are about 1.5 MB max. Most HDDs can't even hit 1 MB/s.
MB/s is actually the inverse of how you perceive speed. You perceive speed as wait time. Less wait time = faster. So if you need to read 10 MB of sequential files from a HDD, @ 150 MB/s it takes just 67 milliseconds - less time than it takes to blink. But if you need to read 10 MB of random small files from a HDD, @ 1 MB/s it takes 10 seconds. So the slower MB/s speeds have a disproportionately larger effect on wait times. In other words, if you want to reduce the time you spend waiting for the computer to do something, you want to improve the slowest thing it does, not the fastest.
So these slow random small file read/write speeds are what contribute the most to your sense that your computer is slow. (Likewise, a SATA 3 SSD gives you something like 85% of the reduction in wait time from a HDD that you'd get from a top of the line PCIe SSD. There's no point paying extra for a PCIe SSD unless you need that little bit of additional wait time reduction.)
SSDs typically hit about 50 MB/s at random small file read/writes (30 MB/s for reads, 70 MB/s for writes). On top of that, the bottleneck actually turns out to be the filesystem interface with the drive, so if you can send the SSD multiple small file requests at once (queuing), it can usually fulfill those requests at 200-500 MB/s.
In other words, a SSD can be hundreds of times faster than a HDD at the tasks which contribute most to your sense that your computer is slow. Boot time is the example which gets trotted out the most, but usually that's only about a 4:1 improvement (from 1 minute to about 15 seconds). Where SSDs really shine is when you're doing lots of random file read/writes.
e.g. If you're doing a virus scan on a HDD, the computer is basically useless. The read/write heads are swinging back and forth at full speed trying to read every file, meaning it can take several seconds to fulfill a new request to read a file. Try to start a game with the scan running, and it can take several minutes longer than without the scan. With a SSD, you can run a virus scan, a malware scan, a chkdsk, run a backup, and compress a folder of small files at the same time, and the computer will still be responsive and usable.