Using Replace in SQL
No, they will not do the same thing.
The amount of effort required by the SQL engine is completely different. In the first query, the engine must go through every row and perform a string replace operation on the Name column. In the second query, it is searching through the table where the name is "Jeff" and just updating the Name column to be Joe.
String replacement is wildcard. So in the first query, the Name "Jeffrey" would become "Joerey".
No, it's not the same.
The first query looks for string within the entire string (see replace() "Replaces all occurrences of a specified string value with another string value."), the second looks for the exact string.
If you have records like JeffJoe, the first query will give JoeJoe, the second query will give JeffJoe (without any modification).