The Romans actually had two alphabets--one for writing on scrolls, notes, etc., and one for carving into stone. The alphabet we all know is the one which was carved on stone, and looks more or less like a variation on that standard computer font, "Times New Roman."
However, the handwritten alphabet--literally "manuscript"--looked somewhat different:
____ ____
/|\ |\ / |\ | | | | | /
/ | \ |/ / | \ | | | |-- /
/ | \ |\ \ | / | | | OR | /
/ | \ |/ \ |/ | | | | /___
A B C D E F F Z
___ ____
| | | | / | |\ /| |\ | / \ | |
|__| | |/ | | \/ | | \| / \ | |
| | | |\ | / | | | \ / |
| | | | \ |/ | | | | | |
H I K L M N O P
____ _____
/ \ |\ / | \ / \ /
| | |/ \ | \ / \/
| | |\ \ | \ / /\
\____\ | \ / | \/ / \
\
Q R S T U/V X
I wish to note that ASCII does not convey curves very well. The manuscript alphabet was not quite this angular, though it was this thin--that is, the "S" looked more like a curved "S" stretched out thin.
As we can see, the main differences can be seen in the way that the letters A, E, O, and P were written. "A" has a vertical middle bar, instead of a horizontal bar; "E" is two parallel vertical lines; "O" is actually more like an upsidedown "U", and "P" is without the closing bottom curve.
The reasons for some of this can be explained as the Roman borrowing from the Greek alphabet: "O" looking like an upsidedown "U" is simply the Greek Ω; "P" is Π. Time and slopping writing/mutations would lead to the Roman scribes curving the lines, leading to a closed space in the letter, as opposed to the open lines. The "A" and "E" are (IIRC) derived from Etruscan letters.
All of this ultimately derives from the Phoenicians of course, but that's another story.