I will install a Linux OS in an old laptop, here is the laptop spec:
Thinkpad T60
2GB ram
ATI Mobility Radeon X1400 (128 MB)
Because there are only 2GB so I wouldn't choose Ubuntu. I have tried Linux Mint and Ubuntu Mate in VirtualBox and I feel they are pretty similar. Which one should I try and why? Thanks!
What Desktop are you using with Mint. Cinnamon or Mate, with only 2g of ram I would use the MATE desktop either on mint or Ubuntu MATE, preferably Ubuntu MATE 
, cinnamon is/ was loosely based on the gnome shell which is what stock Ubuntu uses, and as such tends to use more ram than MATE, meaning less applications you can run before running out of ram or moving to swap(usually pretty slow)
Welcome!
I have pretty much the same T60 laptop with the exception that it has 3GB of RAM.
It has a 32-bit CPU so you will be "limited" to using Ubuntu-based distros that are built on 18.04. It is still a useful laptop with a few caveats:
That said Ubuntu MATE works very well on it and it is able to handle almost anything you'd like to do. That being said mine is used almost exclusively for testing now.
I would recommend burning the ISO's you've downloaded onto a USB and run the laptop off of the Live USB and see how the performance is to you and choose the one that suits you best.
Good luck and have fun.
The Ubuntu Mate with the TraditionalOk theme is the best one suggestion.
Ubuntu Mate is my choice. Why? I tried them all and UM was the best among them. (Shrug)
Hello j2ee
If Ubuntu-Mate works well for you on this machine that will be good news. If you have problems, you might want to consider "MX Linux":
Let us know how you get on. Good luck. 
ubuntu Mate only uses around 500mb ram after start up, I guess it should work well. I will let you guys know how it goes after I get back the old laptop from my parents home!
Ubuntu MATE became pretty greedy for RAM in last releases. In spite of all my love to UM, I recommend Linux Mint with MATE, if you're really want to use MATE DE. If not, better use XFCE or LXDE. Lubuntu with LXDE (not LXQt) is the most lightweight option among Ubuntu-based distros.
As @franksmcb said, the last 32-bit version of Ubuntu MATE will only be supported until 2021. I can't immediately find anything about how long Linux Mint will support 32-bit, but they still have 32-bit current images on their download page, so they might have longer support.
Scratch that: Linux Mint will be dropping 32-bit for version 20. Version 19, based on Ubuntu 18.04. will be supported until 2023, a couple years longer than Ubuntu MATE.
32-bit or 64-bit?
I'm a tad prejudiced, being as I'm on the team for mintCast podcast. But I think either Linux Mint MATE or Ubuntu MATE would be good. The more of a "beginner" you are, the more you should use Linux Mint. I'm not a fan of Cinnamon and use MATE for most things (except Bodhi with Moksha).
The T60 does have a 64Bit compatible Intel core-duo T7200 CPU and is upgradable to 4GBs of RAM you could make a small investment in RAM and use a 64Bit OS such as UM it won't be a speed demon but will function fine