What’s the difference between table.insert(t, i) and t[#t+1] = i

Which to use is a matter of preference and circumstance: as the # length operator was introduced in version 5.1, t[#t+1] = i will not work in Lua 5.0, whereas table.insert has been present since 5.0 and will work in both. On the other hand, t[#t+1] = i uses exclusively language-level operators, wheras table.insert involves a function (which has a slight amount of overhead to look up and call and depends on the table module in the environment).

In the second edition of Programming in Lua (an update of the Lua 5.0-oriented first edition), Roberto Ierusalimschy (the designer of Lua) states that he prefers t[#t+1] = i, as it's more visible.

Also, depending on your use case, the answer may be "neither". See the manual entry on the behavior of the length operator:

If the array has "holes" (that is, nil values between other non-nil values), then #t can be any of the indices that directly precedes a nil value (that is, it may consider any such nil value as the end of the array).

As such, if you're dealing with an array with holes, using either one (table.insert uses the length operator) may "append" your value to a lower index in the array than you want. How you define the size of your array in this scenario is up to you, and, again, depends on preference and circumstance: you can use table.maxn (disappearing in 5.2 but trivial to write), you can keep an n field in the table and update it when necessary, you can wrap the table in a metatable, or you could use another solution that better fits your situation (in a loop, a local tsize in the scope immediately outside the loop will often suffice).