There is an enormous difference. Securityadmin grant server-level permissions to other users. You cannot do that with ALTER ANY LOGIN only.
The demo below illustrates. As the script is given the GRANT CONTROL SERVER command fails. Comment out GRANT ALTER ANY LOGIN and activate ALTER SERVER ROLE instead, and the GRANT fails. Yet, there is no difference in the output from fn_my_permissions.
USE master
go
CREATE LOGIN testsecadmin WITH PASSWORD = 'TeMporrrray'
CREATE LOGIN otherlogin WITH PASSWORD = 'Klklklklklkl'
go
\--ALTER SERVER ROLE securityadmin ADD MEMBER testsecadmin
GRANT ALTER ANY LOGIN TO testsecadmin
go
EXECUTE AS LOGIN = 'testsecadmin'
go
SELECT * FROM sys.fn_my_permissions(NULL, NULL)
GRANT CONTROL SERVER TO otherlogin
go
REVERT
go
DROP LOGIN testsecadmin
DROP LOGIN otherlogin