Is testing Cat5 cable using a testing device necessary if I have verified connectivity with a computer?

These handheld devices are useful, since they test the cabling on the physical level and can help discover problems (cross-talk, wrong impedance, etc.) which are difficult to diagnose otherwise.

If you pulled some cable too much, or some turn inside your wall is too sharp, your network will "kinda work", but you won't get maximum throughput, or you'll get weird connectivity issues, etc.

It will be a pain to re-do stuff once you make everything neat and declare the job is done. Maybe you can lease such a device or hire someone who have it to test your network? These testers are prohibitively expensive to buy if you need it for just one job...

So, it is not strictly necessary to test with such tester, but it gives an extra assurance you did things right, especially if you're not a pro in network cabling.

Yes, you should test it!

The computer may not use all of the pairs... if you're testing with a 100Mbit rather than a Gigabit connection, for example, you're only using two pairs rather than all four. If you don't test this cable properly, it might check out fine on your computer but then fail when you try to use it with a feature like power-over-ethernet, or may only be able to work at 100mbit rather than gigabit speeds.

That said, I often neglect to test my cables. I don't always have a tester to hand, and the cables don't move around that much.