Hi Ken, here's the difference: 【I】 Only "二" can be used: 1) When counting 2) Decimals and fractions 3) To express ordinal numbers 4）After "第" 5) with the tens and hundreds digits 6) in the hundreds, tens and digits of multi-digit numbers 【II】 Only "两" can be used: 1）When used in conjunction with quantifiers 2) directly in front of some nouns as a determiner, but also with other words to form a fixed format. For example: two countries, two schools 3) in front of a unit of measurement, such as: 两吨，两公里 4) To express an approximation, e.g., "去两天" 【III】 In cases where both "二" and "两" can be used: Before "千,万, 亿".

About the numbers： 二 is used to express relatively small numbers，比如，二、十二、二十、二十几（比如23、26、29），几十二（比如42、72、92）、二百、二百几（比如206、230、288）。但是从“千”开始， the number gets bigger, you have to use 两，比如，两千、两千几（比如2400、2997）、两万、两万几。 不过，“二百”in the spoken language is often said as“两百”， this is a special case, just remember it. 例如：22222读作：两万两千二百二十二或者两万两千两百二十二 About serial number： we only use“二”，比如第二名，第二层，第二个 About phone numbers： we also only use“二”，比如13265293321读作幺三二六五二九三三二幺 About number measure words： We usually only use“两”，the structure is：两+measure word(+n.) 比如，两个、两层、两台电脑、两部电影、两块钱

二and两 are two confusing words sometimes even for us Chinese. Simply put: 1. 二 instead of 两 can be used to talk about ordinal numbers: 二叔，二月份，二年级； 2. 二 instead of 两 can be used to talk about numerals: 百分之二，二三十个，二十岁上下，五分之二，一、二、三、四； 3. 两 can be used in front of any measure words: 两本书，两杯水，两个人；二 can only be used in front of some traditional Chinese weights and measures like: 二尺，二里，二亩，二斤. Hope this helps.