CONVERT is SQL Server specific, CAST is ANSI.

CONVERT is more flexible in that you can format dates etc. Other than that, they are pretty much the same. If you don't care about the extended features, use CAST.

EDIT:

As noted by @beruic and @C-F in the comments below, there is possible loss of precision when an implicit conversion is used (that is one where you use neither CAST nor CONVERT). For further information, see CAST and CONVERT and in particular this graphic: SQL Server Data Type Conversion Chart. With this extra information, the original advice still remains the same. Use CAST where possible.

Yes, CAST and CONVERT do the same thing, except that CONVERT allows more options, such as changing character set with USING.

CAST and CONVERT have similar functionality. CONVERT is specific to SQL Server, and allows for a greater breadth of flexibility when converting between date and time values, fractional numbers, and monetary signifiers. CAST is the more ANSI-standard of the two functions. Check this blog for examples of using both of those: http://sqltutorials.blogspot.com/2007/06/sql-cast-and-convert.html

From this link :

1. CAST is an ANSI standard while CONVERT is a specific function in the SQL server. There are also differences when it comes to what a particular function can and cannot do. For example, a CONVERT function can be used for formatting purposes especially for date/time, data type, and money/data type. Meanwhile, CAST is used to remove or reduce format while still converting. Also, CONVERT can stimulate set date format options while CAST cannot do this function.

2. CAST is also the more portable function of the two. It means that the CAST function can be used by many databases. CAST is also less powerful and less flexible than CONVERT. On the other hand, CONVERT allows more flexibility and is the preferred function to use for data, time values, traditional numbers, and money signifiers. CONVERT is also useful in formatting the data’s format.

3. CAST functions also restore the decimals and numerical values to integers while converting. It also can be used to truncate the decimal portion or value of an integer.

No, performance is not a issue.(As far as your question describes both can be used and there is no difference between them but as martin said statement using CAST will cast to varchar(30) as it is it's default length)

CAST is an ANSI SQL-92

CONVERT is specific to SQL Server

CONVERT is specific to SQL Server, and allows for a greater breadth of flexibility when converting between date and time values, fractional numbers, and monetary signifiers.

Try this

SELECT DISTINCT CONVERT(VARCHAR(MAX), GETDATE(),108)
--and the other is saying to use

select distinct cast(GETDATE() as VARCHAR(MAX))