There are MANY changes I would make.
1) Flow, left to right, top to bottom (latin based languages)
This implies that the 'Login' button be placed RIGHT aligned to fields
2) Remove titles
Remove the titles 'Login' and 'Register'. The buttons themselves should act as the title. Make them larger to match.
3) Remove labels
Incorporate the labels INTO the text fields themselves.
4) Combine Login and Register into a single page.
Each additional click a user has to make you lose potentially 20% of your clients. Eliminating clicks either tabs or web pages helps retain more clients.
5) Remove the additional re-type password field
This design element was driven by the engineers of operating systems. Usually an error meant that the user is LOCKED out of the system with no recourse to recover the password. In the web world, the standard way of retrieving your password is via the 'forgotten?' link by email. Also removes extra loops to jump over is always a positive for the user.
6) Move 'remember me' closer to field.
There was some debate whether 'remember me' should be next to the username/email or next to the login/signup. By the field provides it proper context, just after entering his email, he can choose to be 'remembered'. The reasons for placing it next to the 'sign in' button was to also inquire if they were logging in from a SECURE location. Made more sense to the engineers to be placed by the sign in. However it makes more sense from a user perspective that we are remembering his email address (maybe we will also remember his password, maybe not)
7) Missing 'forgotten?'
Again because of its context (next to the password field), its meaning is obvious. The issues of having maybe MIS-TYPED your password in the registration (sign up) panel becomes moot when you have an easy way to recover your passwords.
8) Login or Sign In
In the web world everyone has started to move to a common vernacular, Sign in as opposed to the login of OS based world.
9) Register or Sign Up
If we decide to go with Sign In, then it equally makes sense we would want to go with Sign Up
10) Email or Email address
email is the shorthand for email address, again no need for a more descriptive label
11) Missing Name?
Usually when you register someone you require at least a name. And instead of a first name and last name fields, its best to have a single field and request a full name. Again, note the pattern, we are offloading additional error checking to the programmer. Normally if the programmer designed this he insist on a first name and last name text field which he would directly map into his database.
12) Size and layout.
usually by size you can draw peoples eyes across the login screen. The question is WHERE do you want their eyes to get drawn to.
The trick is NEW users are VALUABLE. Hence you make your design set up to sign up as many new users as possible. That means bigger type, bugger text field, more saturated colors for the REGISTER (or Sign Up) panel.
I quickly threw a mockup together, there is more tweaks, the sign up needs to be even larger, a better separation between signin and signup. Login should use smaller fonts. But the general idea is there.
13) Same fileds for Sign in / Sign Up? - No
Its confusing to have TWO buttons that does TWO different tasks for the user to pick up on. The best way to handle this is to have it managed BEHIND THE SCENES. If a currently registered user tries to sign up INSTEAD of signing in then the software behind the scenes detect that the email is already registered and the password matches, so the system should then AUTOMATICALLY sign in the user and NOT have him retype in all his information again in the sign in panel.