A lot depends upon what type of photos you want to take and how you wish to display them.

Pretty much any camera, including most phones, can take a photo in bright light for relatively low resolution display on the internet. In the hands of the right photographer such an image can even be pretty good no matter what kind of camera is used.

Where the differences are the greatest and the limitations of lesser cameras become more apparent are under specific conditions that tax the light gathering capability of the camera the most, require very wide angle coverage, or challenge the resolving power of sensors and lenses the most.

A camera is a tool. Tools come in different shapes and sizes. Some are better suited for using one way for certain tasks. Other tools are better suited for using other ways to do other tasks.

In general larger sensors allow the possibility of better image quality, but they in no way guarantee it. Sometimes the difference in quality between different cameras taking the same photo will be incremental and hardly even noticeable. Sometimes it will be quite obvious. Often the other things needed to allow using the larger sensor to get that quality means spending a lot more money on things such as lenses in addition to the higher initial cost of the larger sensor.

Where larger sensors have greater advantages:

• The magnification needed to view the image, as projected onto the sensor, at a specific size is smaller for a larger sensor compared to a smaller one. This means the flaws projected by the lens aren't as magnified when the image is viewed. To get the same sharpness with a smaller sensor, the lens must also be sharper.
• Since exposure is measured as the amount of light per unit area, larger sensors collect more total light for the same exposure value. This trends to make larger sensors less noisy. The age of the sensor's technology can also affect this to a significant degree. But if two sensors from the same manufacturer use the same generation of technology the larger one will almost always be less noisy when shooting the same scene with the same settings and then viewing both at the same size.
• Larger sensors tend to have larger photosites which are also known as pixel cells or sensels. Because a larger photosite has more surface area, it can collect more photons before it reaches full well capacity. This gives sensors with larger photosites more dynamic range. DR is the difference between the brightest value a sensor can record and the dimmest value that can be recorded and still be discriminated from noise. This is one reason why larger sensors are less noisy than smaller sensors.
• Since lenses of the same focal length will give a wider angle of view when used with a larger sensor, it is easier to make lenses that give wide angles-of-view for larger sensors.

Where smaller sensors have greater advantages:

• The magnification needed to view the image, as projected onto the sensor, at a specific size is larger for a smaller sensor compared to a larger one. This means a lens with a specific focal length will give greater "reach" with a smaller sensor than a larger sensor. Things the same distance from the camera will appear larger with a smaller sensor than on a larger one when the same focal length is used with each and the respective images are then viewed at the same display size.

• Since aperture is expressed as a ratio of the focal length to the diameter of the entrance pupil, shorter focal length lenses can achieve the same exposure or f-number with a smaller aperture diameter than a longer focal length lens. If a larger sensor needs a 100mm focal length for a certain angle of view and a smaller sensor only needs a 50mm focal length to give the same angle of view, the smaller lens can have an entrance pupil half as wide and give the same f-number as the longer lens.

• Since lenses of the same focal length will give a narrower angle of view when used with a smaller sensor, it is easier and less costly to make telephoto lenses for smaller sensors. The same telephoto lenses will give a narrower angle-of-view (more "zoom") when used with smaller sensors.

In the end, the quality of a tool is only as good as the skill of the one who is using the tool. Only when the tool is limiting the skill set of the user will a better tool make a difference.

For more, please see:

What Do I Gain from Moving to a Full-Frame DSLR?
Why does it seem like large sensors are necessary for good low-light performance?
Do full frame sensors have a higher exposure?
Does sensor size impact the diffraction limit of a lens?
Why are larger sensors better at low light?
What is the visual difference between Full Frame, APS-C and Micro Four Thirds pictures?
When do the differences between APS-C and full frame sensors matter, and why?
Full Frame Vs Crop Frame
Is the low-light advantage of larger sensors attributable to the sensor itself or to the larger aperture of the lenses?
Do I need a full-frame camera for low-light photography?
How does the smaller mirror in APS-C cameras offer these advantages?
What significant improvement will I get if I upgrade to Canon EOS 6D from EOS APSC 600D?
How does sensor size impact depth of field and diffraction for macro photography?
6D or 80D for upgrade from 100D?
Big sensor and landscape photography, DoF
Can we always talk about noise difference between crop and full frame?
Fancy technology cropped vs old technology full frame - Which will give better images?
Is small sensor always a bad thing?
Does sensor size affect lens distortion?
Which sensor "full frame" vs. APS-C (1.6 crop) gives more distortion?
This recent answer to this older question: Are full-frame cameras bad for sports photography?