Kant's theory..

Beautiful 
"This feeling of life brought about by our intuition of an indeterminate object with form is the feeling of pleasure that Kant associates with the beautiful. The feeling of pleasure we get when we encounter a beautiful object, says Kant, is nothing other than “a feeling of the free play of the powers of representation in a given representation for cognition in general.” (Kant, p. 102) When we judge an object in a merely reflecting way, we feel pleased in the awareness of the “animation of both faculties (the imagination and the understanding).” (Kant, p. 103) This animation is what makes us judge the object to be beautiful. For Kant, an object is considered beautiful if and only if it is an indeterminate self-enclosed object, which allows itself to be contemplated freely, without any constraint from empirical or ethical concepts.

Thus the beautiful, for Kant, is neither a feeling of agreeableness in the senses (the empiricist model), nor the perception of perfection in an object (the rationalist model). The beautiful, for Kant, is the result of a merely reflecting judgment which, in refusing to attribute any determinate content to the object at hand, enables a pleasurable mode of thinking, where the imagination and the understanding are in a free and harmonious relation to one another." - http://courses.washington.edu/asthetik/thiti_beautiful.html

A persons imagination and what they see of something through that imagination is what allows things to be seen as beautiful. It is not about the perception of an object or how it actually looks but more about what an individual persons mind and imagination allows them to see. This is what allows us to see and determine that things are beautiful. But everyones view of what is beautiful will be different due to what their imagination allows them to see, again no two people will be able to imagine the same thing because imagination if fueled by personality and experiences that have taken place.


Sublime
"Like the judgment of the beautiful, the judgment of the sublime, for Kant, is a type of merely reflecting judgment. As such, these two forms of judgment share one important feature with one another: they both give rise to pure aesthetic pleasure – i.e. pleasure not grounded in any determinate empirical or ethical concept. Just as the pleasure of the beautiful is the result of a certain mode of thinking, rather than any particular attribute of the intuited object, so is the pleasure of the sublime the result of a certain process of thought, which is unrelated to any objective feature of the experienced phenomenon."

"the sublime is to be found only in a phenomenon of experience that is boundless, without any discernible shape within its spatial or temporal structure"

http://courses.washington.edu/asthetik/thiti_sublime.html

In Kant's theory the sublime and the beautiful are similar in respect to how his judgments differ.
As each one person has a different view of life which is shaped by experiences that they have had what they see as sublime will be different from the next person. But the general feeling of the sublime is about things being exceptional and going above what could have ever been though to be possible. Things being great and beyond what is see as normal, the unexpected things in life.  I think what Kan't is getting at with the term sublime is things being normal but things that we don't really have to think of as being normal, the everyday things that we pay no attention too as they are familiar and a way of life, he wants to go av=o


Mathamatical Sublime

" Now that we have a general sense of what Kant means by the sublime, it is time to discuss the more specific notion of the mathematical sublime. The mathematical sublime is a feeling of the sublime which we experience when we encounter something overwhelming in size. Thus, to understand what gives rise to the feeling of the mathematical sublime, we need to understand how Kant conceives of the mind to be capable of judging sizes.
Kant makes a distinction between two ways of estimating size: aesthetically and mathematically. An aesthetic estimation of size, says Kant, is something that occurs “in mere intuition (measured by eye).” (p. 134) When we run into a tall person on our way to the office, for example, we judge the person to be tall, not through the use of a ruler, but through what we consider to be the norm of a person’s size – i.e. through “the average magnitude of the people known to us.” (p. 133) An aesthetic estimation of size is thus something that we do every day, intuitively, without any attempt at calculation.

A mathematical estimation of size, on the other hand, is something that requires us to calculate (i.e. brings things into numerical relations with one another). When we want to judge whether a person’s height has broken the world record, for example, we do not simply eyeball the person’s height: we measure the person’s height quantitatively, “by means of numerical concepts.” (p. 134) Thus, in a mathematical estimation of size, instead of using the norms of intuition as our guide, we use the norms of reason (i.e. numerical relation) for our estimation. A mathematical estimation of size is thus an estimation of size based on quantitative reasoning, which requires us to compare the size of things based on their numerical relations to one another, rather than judging merely on the basis of our intuitive norms"  - http://courses.washington.edu/asthetik/thiti_mathematical.html  

Mathematical sublime differs from the sublime as it needs to be thought about more, it goes above the normal every day things that are seen and when we have to think about certain things and try and calculate their significance and what they mean.