Most of us use these two terms interchangeably and thus do not have a clear idea about the difference between the two.
Dictionary meaning of feelings says :
* something that you feel in your mind or body eg. happiness, hunger, fear, helplessness
* a belief or idea that something is true or is likely to happen eg. get a feeling that he doesn't like me
* an attitude or opinion about something eg. what is your feeling in this matter?
* love or understanding for somebody/something eg. He still feels for his ex-wife or doesn't have feeling for music
* the ability to feel in your body eg. I don't feel any pain in my leg

We will find that feelings are more related to mind's state or deciphering from inputs out of the five senses

Now let us see what is the dictionary meaning of emotions say :

Wikipedia : Emotions are mental states brought on by neurophysiological changes, variously associated with thoughts, feelings, behavioral responses, and a degree of pleasure or displeasure.

There is currently no scientific consensus on a definition. Emotions are often intertwined with mood, temperament, personality, disposition, or creativity. Emotions are not causal forces but simply syndromes of components, which might include motivation, feeling, behaviour, and physiological changes, but none of these components is the emotion. Nor is the emotion an entity that causes these components.

According to Scherer's Component Process Model (CPM) of emotion, there are five crucial elements of emotion.
1. Cognitive appraisal: provides an evaluation of events and objects.
2. Bodily symptoms: the physiological component of emotional experience.
3. Action tendencies: a motivational component for the preparation and direction of motor responses.
4. Expression: facial and vocal expression almost always accompanies an emotional state to communicate reaction and intention of actions.
5. Feelings: the subjective experience of emotional state once it has occurred

Dictionery : a strong feeling such as love or anger, or strong feelings in general
eg. Like a lot of men, he finds it hard to express his emotions or My mother was overcome with emotion and burst into tears.
Dr. James Gordon (founder of the Center for Mind-Body Medicine) says "the brain and peripheral nervous system, the endocrine and immune systems, and indeed, all the organs of our body and all the emotional responses we have, share a common chemical language and are constantly communicating with one another."

Here we find that emotions are mostly external expressions of the state of mind.
Feelings as a state of mind
Lets see what remains in mind if we take out the other elements which come as an input to the mind. Which are the input sources
External Senses through brain
Internal senses through brain
Memories through brain
Processed data through brain, etc
Let's stop all these flows. All of these are related to the body as brain is part of body. Sensors are connected to brain processing and brain memory unit. Memory is part of brain. With body gone the brain is gone and we will not have the senses, memory and processing power. So the state of mind remains static with only the state of happiness or unhappiness, love or hate, ego or egoless feelings. This state remains static and unable to express. As for the expression of these, one needs a medium i.e. brain and body.
These feelings as it remains as the residual elements of the mind, is also a indication of our karma phal and also the reflected consciousness.
We talk about bliss state and unconditional love as the basic state of us - Sat Chit, Anand. This is that static permanent feeling of mind and beyond mind.
At death we leave our body, thus we leave our memory, our sensors, our processing power and whatever are physical or the gross elements.
What we carry are residual state of the feelings and ego which again is related to the reflected consciousness, the drivers for the sensors and other body elements and the subtle intellect which is the intelligence behind the entire functioning.
Now which are the different scenarios which evoke emotional responses ?
These can be many as emotions are driven by lot more factors other than state of mind. And that is why the emotional expressions vary across different persons. Some are more emotional and some are less; some are emotional in certain aspects, some are not; again, some have longer or lingering emotional state, some have very short emotional state; in general females are more emotional, males are less.
I will try to bring out some aspects of the causes to these. Quite a thought-provoking subject as the understanding these might help in appreciating our relationships better.
Emotions as a direct link to feelings – The passion of emotional expression varies. The feelings can be love, happiness or the absence of these. As the feelings of love and happiness varies due to the other factors which cloud them, the emotional expression varies. As mentioned earlier, our basic nature is love and being happy. More are the layers of cloud, more we are away from our basic nature. More away from love and happiness.
Gap in the expectations and reality – This is most often the cause of our emotional expressions. As we try to derive our happiness and love from external sources, our dependencies increase and more often than not we are not satisfied and thus we get emotional. More so if we do not state our expectations upfront and thus leave to the intuition and imagination of others to satisfy me. Many a times this can be an one off matter and the relationship can break, though there might be hundreds other situations where it might have been fine or even better resulting in happiness.
Mental state due to other reasons – This may not be related to any related expectations but related totally to other factors like body, mind state (due to other external factors ), etc. But the impact can come on the related relationship. For example one can have body pain or is down due to some adverse situation of loved ones or even situations. In this case it can influence the emotional factor adversely.
There are other side also. The positive emotions which can be a result of the positive aspects of the above.
Females are generally more emotional than males. One purpose and design of females is to bear and rear children. Also they are mostly designed to take care of home and families. So managing relationships is of utmost importance for them. These are where the emotional factors come into picture. Emotions connect people much faster than anything else. Woman’s tears are part of this, our homes. In our tradition they are the lakshmi of home. They lighten up the homes. These are never said about males. Their empathy, tears to express their emotions, etc are unparallel. However in this process of managing relationships through the extreme complex web of expectations, they also have to keep many things in mind – un-expressed, unsaid. Sometimes these create the gaps
Another aspect which is often seen is the time duration of the lingering negative emotions. When is the time duration long ? Say suppose a person Y has not been able to meet the other person’s (X) expectations out of love. It can be due to many reasons. This led to emotional outburst and subsequent phase of no communication or cold war. This phase can be from few hours to months or more. Now this brings the question about love. Whether person X loves person Y more than her emotions or expectations. Can love be without expectations ? Are lingering negative emotions are related to higher ego ? i.e. ego is hurt when one is loving one’s emotions and expectations more than anything else. This is a common scenario but is quite complex if we go into the practicalities. Ideally love should be beyond expectations but in reality this is difficult to achieve. Thus we find breaking of relationships or long standing emotionally disconnected and cold war situation. From transactional world of relationships point of view, these pulls and pushes might be necessary to bring more balance in the relationships but in absolute terms these might lead to more ego nurturing and more possessiveness of one’s likes and dislikes.

Thus the feelings which are in mind get expressed in different forms of emotions depending on the situations, person’s nature, body composition, etc.
To judge one’s feelings, the emotional quotient might not be the only true index but it needs to be judged through a number of parameters of actions over a time period for more nuanced understanding. Else chances are there of misunderstanding and mis-judgment thus resulting in possible wrong actions.