Night and day. The differences can easily fill a chapter of a book, but here is a high level summary of some of the differences.
Here is an example of code of an ASP page. You'll notice that ASP code is within the <% and %> tags.
<title>My ASP Page</title>
Dim oConn, oRS, ds, sql, oCmd, qs
ds = "Provider=Microsoft.ACE.OLEDB.12.0;Data Source=C:\inetpub\wwwroot\test.mdb;Persist Security Info=False;"
sql = "SELECT * FROM Employees where empID = ?"
qs = request.QueryString("qs")
Set oConn = Server.CreateObject("ADODB.Connection")
Set oCmd = Server.CreateObject("ADODB.Command")
oCmd.ActiveConnection = oConn
oCmd.CommandText = sql
oCmd.CommandType = 1
oCmd.Prepared = True
oCmd.Parameters(0) = qs
Set oRS = Server.CreateObject("ADODB.Recordset")
do until oRS.EOF
for each x in oRS.Fields
Response.Write(x.name & "=" & x.value & "<br />")
ASP.NET is object oriented, uses the event-driven model, where all your code executes in response to an event, such as a "Page Load", "Button Click", etc. You can use many different languages to write your code: Visual Basic, C#, J#, etc. All code is compiled to use the same Common Language Runtime (CLR) that resides in the .NET Framework. Very similar to the concept in Java. Variables are strongly typed, and the same scoping rules that apply in any recent, major programming language.
Here is a very simple example of an asp.net page written in VB.NET which simple takes information from the Server Variables collection and during the page load event, assigns that information to a label control.
<%@ Page Language="VB" %>
Protected Sub Page_Load(sender As Object, e As System.EventArgs) Handles Me.Load
lblIP.Text = Request.ServerVariables("remote_addr")
Page.Title = "My Page"
<form id="form1" runat="server">
<asp:Label id="lblIP" runat="server"></asp:Label>
There is no requirement to learn ASP before you learn ASP.NET. You should pick up some books on ASP.NET, and download Visual Studio from the Micorosft site and start practicing.
As per the ASP and ASP.net difference concern i like to say that ASP is an interpreted programming language and ASP.net is compiled programming language. ASP use the technology called ADO where ASP.net use the ADO.net technology. ASP is the partially object oriented programming language where the ASP.net is the purely object oriented programming language.
In classic ASP there was no server controls. You have to write all html tags manually. ASP.NET offers a very rich set of controls called Server Controls and Html Controls. It is very easy to drag and drop any controls to a web form. The VS.NET will automatically write the required HTML tags automatically for you.