You can but synaptic is easier as it has been tested for compatibility. But go to the source of any software you can get through synaptic and you will get the source code which has to be compiled (the plus side is a far smaller and often faster package!) or there are binaries with lots of larger packages. The problem is, as you will see, that every software provider has to make separate executables for every type of distro (.deb, rpm's, yum etc etc).
I can't imagine what benefit you can see. The windows way includes either buying your software or getting a compiled version of what is available to linux users. Some .exe files work with Linux via wine or crossover office etc. If you would like to see more screen shots etc then you will find that most apps in synaptic have either their own sites with screenshots or are reviewed somewhere with screen shots or are hosted for download via sourceforge but there are some here without screenshots.
You say you want to search for it on the internet, download it and install it. Well that is exactly what synaptic does. Ubuntu uses the apt package manager. So if you use the terminal to install firefox, say, you would type:-
sudo apt-get install firefox
Synaptic is just the graphical user interface that this process so that you can use a mouse and just point and click.
Here are some places where you can go for software
http://www.getdeb.net/
http://linuxappfinder.com/
http://www.debian.org/distrib/packages
http://linux.softpedia.com/
http://sourceforge.net/
There are lots of others but you must be careful what you install. You don't sound like you have a lot of experience and fixing a broken package, though easy, is difficult to explain. I think a lot of people forget that MS Windows was about for a very long time and to begin with didn't have much functionality. This made it easier to learn. People picked it up over a decade and a half AND it is crap. Linux is a little more difficult to learn but has far greater power, functionality, maintainability and can be made to do so much more than a windows box. You are in charge. Not microsoft.
I have a program on my laptop that knows where my bluetooth enabled devices, namely a phone, are. The laptop will lock down and go to a screen saver if I walk more than, say 5 metres away. When I return it comes back on or comes back on with the password. A visitor can leave a message on the screen but doesn't have access. When I leave or return it can start one or several programs. I don't have to be able to program. I just put the distances and the names of the programs plus some options. It could just hibernate or standby if I never went more than 50 metres away (ie the bathroom) or it could start a virus check if I left the building. That's just one dumb program that no one knows about but I've had it for 2 years.
If you want to open firefox at getdeb.net you just need to type:
Alt+F2
firefox getdeb.net
What I am saying is that you don't need another Linux distro to that because Ubuntu does it already BUT the downside is you may have to type three or four commands to install the package if it is not compiled for you. The commands which you type in a terminal (Alt+F2 and type gnome-terminal and ENTER) are typically:-
# tar xvzf package.tar.gz (or tar xvjf package.tar.bz2)
# cd package
# ./configure
# make
# make install
sometimes they are little different but there is always a readme file with the download - if the first command didn't work it is because you are in the wrong place or you downloaded the package to the wrong directory or folder. Download the uncompiled package to what is printed when you type:-
pwd
and hit ENTER in a terminal and the above will work fine
OR type:-
cd /home/yrjokin/Desktop
that would be if you downloaded it to the 'Desktop' AND your name was 'yrjokin'
by the way: where it says package in the above instructions you type the name of the application but you can often finish the whole title by hitting 'TAB' so long as there is nothing else with the same letters so the computer knows which one you mean. The same applies to any command but as there are literally a few thousand it does often get confused with command names. Type fir and hit TAB - firefox will print out - now hit enter or type the web address or as little of it as you can get away with!
Okay, I think that is it. Why you do not want to use Synaptic or the terminal is a mystery to me but at least not you can use gDebi to install packages or you can compile from source code if you get that. You may have to install a program called gcc from synaptic if you want to build packages from source. Read the output from the terminal :-
looking for 'gcc' : No (or something similar (ish) )
gfortran is sometimes used too so install it and ubuntu-restricted-extras (only if you are using gnome (it is xubuntu-restricted-extras for xfce or kubuntu-restricted-extras for KDE)) these are codecs and the flash player for You Tube etc