In PostgreSQL, I use the now() and current_timestamp function and I see no difference:

# SELECT now(), current_timestamp;
              now               |              now               
--------------------------------+--------------------------------
 04/20/2014 19:44:27.215557 EDT | 04/20/2014 19:44:27.215557 EDT
(1 row)


Am I missing something?

There is no difference. Three quotes from the manual:
1.

These SQL-standard functions all return values based on the start time
of the current transaction:
...
CURRENT_TIMESTAMP
...





transaction_timestamp() is equivalent to CURRENT_TIMESTAMP, but is
named to clearly reflect what it returns.





now() is a traditional PostgreSQL equivalent to transaction_timestamp().

Bold emphasis mine. CURRENT_TIMESTAMP, transaction_timestamp() and now() do exactly the same. CURRENT_TIMESTAMP is a syntactical oddity for a function, having no trailing pair of parentheses. That's according to the SQL standard.
If you don't declare a column alias for a function call in an SQL statement, the alias defaults to the name of the function. Internally, the standard-SQL CURRENT_TIMESTAMP is implemented with now(). Up to Postgres 9.6 that shows in the resulting column name, which was "now", but changed to "current_timestamp" in Postgres 10.
transaction_timestamp() does the same, but this one is a proper Postgres function, so the default alias has always been "transaction_timestamp".
Do not confuse either of these functions with the special input constant 'now'. That's just one of several notational shorthands for specific date/time/timestamp values, quoting the manual:

...  that will be converted to ordinary date/time values when read. (In particular, now and related strings are converted to a specific time value as soon as they are read.) All of these values need to be enclosed in single quotes when used as constants in SQL commands.

It may add to the confusion that (up to at least Postgres 12) any number of leading and trailing spaces and brackets ({[( )]}) are trimmed from those special input values. So 'now()'::timestamptz - or just 'now()' where no explicit type cast is required - is also valid and happens to evaluate to the same timestamp as the function now() in most contexts. But those are constants and typically not what you want as column default for instance.
db<>fiddle here
Old sqlfiddle
Notable alternatives are statement_timestamp() and clock_timestamp(). The manual:

statement_timestamp() returns the start time of the current
statement (more specifically, the time of receipt of the latest
command message from the client). [...]
clock_timestamp() returns the actual current time, and therefore its
value changes even within a single SQL command.

Note: statement_timestamp() is STABLE as the above (always returns the same value within the same SQL command). But clock_timestamp() necessarily is only VOLATILE. The difference may be significant.
