Consonants are distinguished normally by features like place of articulation, manner of articulation, voiced/voiceless, etc. while vowels are usually distingusihed by stuff like tongue's position and roundedness. Why can't a universal methodology be used to classify both of them? Some of them are similar anyway, e.g. /i/ vs /j/
Because that is precisely what separates consonants from vowels: consonants are sounds produced with a partial closure of the vocal tract. Depending on what type of closure the speaker does (in combination with other factors), a different consonant will be pronounced. And, as you have already seen, consonants can be analyzed according to several features: place of articulation, manner of articulation, aspiration etc.
Vowels, on the other hand, are pronounced with an open vocal tract. Therefore, by definition, it is not possible to apply those features to classify vowels. Other features, such as roundedness, must be used in this case. So, for example, it wouldn't make sense to speak of place of articulation for vowels.
Basically, vowels are syllable nuclei, and consonants are syllable peripheries. Consonants are the sounds that don't occur in the middle of a syllable, and vowels are the ones that do.
That's all, really.
Aside from diphthongs like /ay/ in light, which involve tongue movement during pronunciation, vowels are determined by the shape of the resonating chambers formed by the tongue in the mouth (this is what decides the formants).
Consonants, however, are always modulated by approach or touch of an articulator (usually some part of the tongue) to another part of the vocal chamber, and they have very complex wave-forms, with fricatives being especially noisy.
Independent articulators outside the inner mouth cavity can apply to either kind of sound because they don't participate in this alternation. For example, the lungs are independent and therefore there are both aspirated vowels and aspirated consonants; the lips can round vowels or consonants; the larynx can glottalize vowels or consonants, as well as voice either, etc.
The SPE feature system is actually the "universal methodology" that you seek. This is because the features represent independently controllable aspects of articulation. In his book Speech Sounds and Features, Gunnar Fant called this property of the feature system "orthogonality". It means that for the binary features, all combinations of + - are possible, so that 2^(number of features) different sound segments can be distinguished.
The appropriate feature difference between "consonants" and "vowels" is not terribly clear, but let's say vowels are [+vocalic] and consonants are [-vocalic]. Then the number of possible vowels is the same as the number of possible consonants, and for the consonants classified by the place of articulation features, there are also vowels classified in exactly the same way, by place of articulation.
Chomsky and Halle do depart from the orthogonality assumption when they declare the combination [+high,+low] to be impossible.
The first person to make theoretical use of feature orthogonality, so far as I know, was William Holder in Elements of Speech. Holder argued that there must be a velar nasal consonant in the series [m, n, ...] to match the places of articulation of [p, t, k].
While others have provided informative answers to this question, I would argue that the premise of this question is not completely accurate.
For example, in the traditional distinctive feature set, all phonemes (both vowels and consonants) are specified for the features [cons] and [son]. Further, it's true that there are additional features for which only vowels or only consonants are specified, but in a way this is an arbitrary distinction to single out. For example, in the same way that all vowels and sonorant consonants and glides are trivially [+son], all consonants besides /h/ and /ʔ/ are trivially considered [-spread] and [-constr], meaning those two features are not used to distinguish a vast majority of consonants. The resulting effect is that the minus value for those features on most consonants doesn't amount to much more than a placeholder. One could easily assign a minus values for those features on vowels and it would bear the same significance.
It's true that some of the features used to distinguish vowels appear to have redundant analogues in the set used to distinguish consonants (e.g. [rounded] and [labial]), but really that is just a side effect of the fact that they are derived from similar articulatory properties. It is important to keep in mind that the distinctive features are just a convenient way to categorize phonemes at the phonological level (which is why I added the phonology tag to the post) and at the end of the day they are abstract labels derived mnemonically from articulatory (and in some cases acoustic) characteristics of the speech units with which they are associated.
It has been known for some time that there is huge mismatch between the terminology and the reality when we talk about the description of vowels. Vowels are rather simple acoustically (they can be pretty well described by their formant structures), but rather complex in the articulatory domain (articulations are continuous, and different articulations can give rise to similar formant effects; for instance, both lip rounding and backness depress F2 and are in a sense contradictory—that’s why languages usually have [u] and [i] and much more rarely [y] and [ɯ]). So, when we are talking about vowels we are effectively describing their sounds (and use the formant plots for studying vowel systems). Consonants, on the other hand, are very complex acoustically because they mostly consist of aperiodic turbulent noise, which is hard to describe in a concise fashion; however, it often easy to describe the positions of different articulators because those are mostly well defined and ‘discrete’ (tongue pressed against teeth or the alveolar ridge, etc.) and this is often done using different instrumental techniques like palatography and ultrasound. Therefore, when describing consonants and vowels, we not only use different words, but also actually mean completely different things by them so a chance for a unified descriptive methodology is pretty slight.
A universal methodology can be used to put together vowels and consonants and indeed you will find this occasionally, mostly in phonetics though - e.g. IPA can be, in a way, considered one such methodology with totally closed occlusives through nasals, fricatives up to the approximants with highest degree of aperture at the very bottom. This is follow by a vowel chart typically placed beneath the consonants, again with the most closed vowels (following directly the approximants) at the top and the most open vowels at the bottom (it is rather ingenious, really). Another method is accoustic description (formants in vowels and approximants/liquids, noise formants in fricatives and formant loci in occlusives).
The problem is there will be almost no overlaps between the two (except the /i/ and /j/, /u/ and /w/ and several other examples) in phonology because they are articulated in a wildly different manner and have wildly different position with the phonological system (e.g. with regards to combinatorics) due to their phonetic properties. Thus you will get just one large/wide table/tree instead of two smaller but equally descriptive. It is simply not useful to treat vowels and consonants in a single system.
Every generative theory of phonology using features has assumed that consonants and vowels are described with the same features, except for a brief period in the 80's under the Steridian theory which had a segregation of vowel tongue body and consonant tongue body (Velars vs. Dorsals). SPE half-failed to reach the desideratum of full orthogonality, but head-nodded in the direction of [syllabic], leading to the elimination of the last bastion of the autonomous vowel - consonant distinction, and the field adopted [syllabic] the next day. The question presupposes that there is a well-defined distinction between "consonant" and "vowel", which is exactly what post-SPE theories and observation of language denies; and thus long debates over high vowels versus glides adjacent to the segment which is the syllable peak, the problem of syllabic consonants like /m̩ r̩/ and even syllabic obstruents in Berber, the problematic non-consonantal consonants [h ʔ].
Part of the question is ambiguous: "Consonants are distinguished normally by features like place of articulation, manner of articulation, voiced/voiceless, etc. while vowels are usually distinguished by stuff like tongue's position and roundedness", which could refer to the underlying analytical theory ("why do we distinguish them...."), or it could refer to the facts themselves (e.g. why is voicing not usually contrastive on vowels). Especially if the intent is to focus on apparent disjointness between "consonants" and "vowels", it's important to clarify what things count as "consonants" versus "vowels".