Well...you guys started off pretty hard. That's not a common trait of muay thai, at least not traditionally. For being your first fight it's really not bad. Like most guys just starting out--it's frantic and a little uncontrolled, but that's because both individuals were sort of in survival mode straight from the get-go. The subconscious fear of not knowing what's to come pumps an overproduction of chemicals and you end up gassing sooner partly because of that as well.
In general, work on maintaining a proper guard and cleaning up your punching technique. You flail more than you actually punch and you leave a lot of holes in your guard. While the elbows are held out a bit further for muay thai, the hands need to still be by your face and not out to the sides. At least in the traditional sense.
I really like to see clinch work mixed in with Muay Thai. It's what separates the men from the boys in my opinion. The clinch work you both focused on was the inside dominant grip in an effort to throw knees, but remember that you can throw knees from just about any clinch position and really within punching range as well. Maintain control of the head and arms...suck up the space and only give space when YOU want to knee. Time it, don't just constantly knee like you would trading haymakers. You caught one of his knees at around the :55 marker, that would have been a good time to have dumped him there by twisting to the same side as his caught knee. Also, at 1:09 he had your knee and you used one of the standard clinch defenses, but a little trick that you can use also is is to angle your knee a little more like a half shin-half knee kick, and put pressure on that side as you pull with your left arm to twist while hopping on your standing leg a little over to your right. This should twist him over to your left and as you rotate around you can land on top of him. This looks better to judges because you're the one on top.
That said, I'd also like to have seen you time the guy with jabs and teeps first to let him know that you can read him. It's a great way to diminish his confidence when he gets hit every time he tries to his you. As far as other kicks go...you didn't really have a lot of heat on them. You seemed to kick with the only intention to get into punching/clinching range. That's a valid tactic but think about how much more effective you'd be if the kick hurt him too? I know with the pads though you basically don't even feel the kicks, but it's the idea behind it.
Not too bad brotha. Good job for your first time out. It doesn't get easier as time goes on, but it does get better and more controlled as you get used to fighting. Just wait till you do it for the first time without shin guards on...a whole new world then because you can drop a guy with a single good kick.