What you've got are an ordering query, TOP, and an aggregate query, MIN. The ordering query must have an ORDER BY statement to work. You can't know what order things will come out of the db in. So, to compare these two, you'd need to write the first one like this:

SELECT TOP 1 * FROM TicketActivity

ORDER BY TicketID ASC

The ORDER BY statement will establish the order of the records returned and the ASC will make them ascend from the lowest to the highest. So this query will return the same value as the other.

BTW, depending on your indexes, this query is likely to perform better than the other. The reason for this is that ordering the values from an index is a pretty inexpensive operation, but aggregating values is usually more expensive. But your mileage may vary, so test within your environment.